Partially Ordered Rings and Semi-Algebraic Geometry (London Mathematical Society Lecture Note Series 37)

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: PROFESSOR I.M. James, Mathematical Institute, 24-29 St.Giles, Oxford Already published in this series 1. 4. 5. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

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General cohomology theory and K-theory, PETER HILTON. Algebraic topology: A student's guide, J.F. ADAMS. Commutative algebra, J.T. KNIGHT. Introduction to combinatory logic, J.R. HINDLEY, B. LERCHER and J.P. SELDIN. Integration and harmonic analysis on compact groups, R.E. EDWARDS. Elliptic functions and elliptic curves, PATRICK DU VAL. Numerical ranges II, F.F. BONSALL and J. DUNCAN. New developments in topology, G. SEGAL (ed.). Symposium on complex analysis Canterbury, 1973, J. CLUNIE and W.K. HAYMAN (eds.). Combinatorics, Proceedings of the British combinatorial conference 1973, T.P. McDONOUGH and V.C. MAVRON (eds.). Analytic theory of abelian varieties, H.P.F. SWINNERTONDYER. An introduction to topoligical groups, P.J. HIGGINS. Topics in finite groups, TERENCE M. GAGEN. Differentiable germs and catastrophes, THEODOR BROCKER and L. LANDER. A geometric approach to homology theory, S. BUONCRISTIANO, C.P. ROURKE and B.J. SANDERSON. Graph theory, coding theory and block designs, P.J. CAMERON and J.H. VAN LINT. Sheaf theory, B.R. TENNISON. Automatic continuity of linear operators, ALLAN M. SINCLAIR. Presentations of groups, D.L. JOHNSON. Parallelisms of complete designs, PETER J. CAMERON. The topology of Stiefel manifolds, I.M. JAMES. Lie groups and compact groups, J.F. PRICE. Transformation groups: Proceedings of the conference in the University of Newcastle upon Tyne, August 1976, CZES KOSNIOWSKI. Skew field constructions, P.M. COHN. Brownian motion, Hardy spaces and bounded mean oscillation, K.E. PETERSEN. Pontryagin duality and the structure of locally compact abelian groups, SIDNEY A. MORRIS. Interaction models, N.L. BIGGS. Continuous crossed products and type III von Neumann algebras, A. VAN DAELE. Uniform algebras and Jensen measures, T.W. GAMELIN. Permutation groups and combinatorial structures. N.L. BIGGS and A.T. WHITE. Representation theory of Lie groups, M.F. ATIYAH. Trace ideals and their applications, BARRY SIMON. Homological group theory, edited by C.T.C. WALL.

Partially Ordered Rings and Semi-Algebraic Geometry

Gregory W. Brumfiel

CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE LONDON NEW YORK NEW ROCHELLE MELBOURNE SYDNEY

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521228459 © Cambridge University Press 1979 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1979 Re-issued in this digitally printed version 2007 A catalogue record for this publication is available from the British Library ISBN 978-0-521-22845-9 paperback

Contents

Page INTRODUCTION

1

CHAPTER I - PARTIALLY ORDERED RINGS 1.1. Definitions

32

1.2. Existence of Orders

33

1.3. Extension and Contraction of Orders

34

1.4. Simple refinements of orders

36

1.5. Remarks on the Categories (PORNN) and (PORCK)

37

1.6. Remarks on Integral Domains

39

1.7. Some Examples

40

CHAPTER II - HOMOMORPIIISMS AND CONVEX IDEALS 2.1. Convex Ideals and Quotient Rings

45

2.2. Convex Hulls

46

2.3. Maximal Convex Ideals and Prime Convex Ideals

49

2.4. Relation between Convex Ideals in

(A,'p) and

(A/I,'IV I) ... 52

2.5. Absolutely Convex Ideals

52

2.6. Semi-Noetherian Rings

,

56

2.7. Convex Ideals and Intersections of Orders

62

2.8. Some Examples

66

CHAPTER m

- LOCALIZATION

3.1. Partial Orders on Localized Rings

77

3.2. Sufficiency of Positive Multiplicative Sets

79

3.3. Refinements of an Order Induced by Certain Localizations .. 80 3.4. Convex Ideals in

(A/J3)

and

(A T ,^ T )

81

3.5. Concave Multiplicative Sets 3.6. The Shadow of 1

83 ;

84

3.7. Localization at a Prime Convex Ideal

87

3.8. Localization in (PORCK)

88

3.9. Applications of Localization, I - Some Properties of Convex Prime Ideals

89

3.10. Applications of Localization, H-Zero Divisors

91

3.11. Applications of Localization, m - Minimal Primes, Isolated Sets of Primes, and Associated Invariants 3.12. Operators on the Set of Orders on a Ring

93 96

CHAPTER IV - SOME CATEGORICAL NOTIONS 4.1. Fibre Products

101

4.2. Fibre Sums

102

4.3. Direct and Inverse Limits

103

4.4. Some Examples

104

CHAPTER V - THE PRIME CONVEX IDEAL SPECTRUM 5.1. The Zariski Topology Defined

106

5.2. Some Topological Properties 5.3. Irreducible Closed Sets in 5.4. Spec(A,'p)

107 Spec(A,'£)

107

as a Functor

5.5. Disconnectedness of

109

Spec(A,'}3)

109

5.6. The Structure Sheaf, I - A First Approximation on Basic Open Sets

112

5.7. The Structure Sheaf, II - The Sheaf Axioms for Basic Open Sets

113

5.8. The Structure Sheaf, HI - Definition

115

CHAPTER VI - POLYNOMIALS 6.1. Polynomials as Functions

118

6.2. Adjoining Roots

120

6.3. A Universal Bound on the Roots of Polynomials

123

6.4. A "Going-Up" Theorem for Semi-Integral Extensions

125

CHAPTER VII - ORDERED FIELDS 7.1. Basic Results

130

7.2. Function Theoretic Properties of Polynomials

132

7.3. Sturm's Theorem

135

7.4. Dedekind Cuts; Archimedean and Non-Archimedean Extensions. 137 7.5. Orders on Simple Field Extensions

140

7.6. Total Orders and Signed Places

144

7.7. Existence of Signed Places . . ..

148

CHAPTER VIII - AFFINE SEMT-ALGEBRAIC SETS 8.1. Introduction and Notation

i62

8.2. Some Properties of RHJ-Algebras

I68

8.3. Real Curves

,

178

8.4. Signed Places on Function Fields

184

8.5. Characterization of Non-Negative Functions

193

8.6. Derived Orders

196

8.7. A Preliminary Inverse Function Theorem

206

8.8. Algebraic Simple Points, Dimension, Codimension and Rank . 212 8.9. Stratification of Semi-Algebraic Sets

218

8.10. Krull Dimension

224

8.11. Orders on Function Fields

232

8.12. Discussion of Total Orders on

R(x,y)

240

5.13. Brief Discussion of Structure Sheaves

247

I - The rational structure sheaf

248

II - The semi-algebraic structure sheaf

252

IH - The smooth structure sheaf

262

APPENDIX.

268

The Tarski-Seidenberg Theorem

BIBLIOGRAPHY

273

LIST OF NOTATION

278

INDEX

279

Preface

This text represents an attempt to formulate foundations and rudimentary results of a type of geometry and topology in purely algebraic terms.

I feel

that the approach taken here is very natural and that it is only coincidental that the point of view I advocate did not emerge fifty years ago. The mathematics itself is most similar to elementary commutative algebra and algebraic geometry.

The level of difficulty is about like that of the

texts on commutative algebra by Zariski - Samuel or Atiyah-Macdonald.

Although,

strictly speaking, the text might be read without any previous knowledge of basic commutative algebra, essential motivation would probably be lacking. On the other hand, I see no reason why a student couldn't simultaneously read this text and some classical commutative algebra. In the final two chapters, I assume the reader is familiar with, or can read elsewhere, basic results of Artin-Schreier theory, Krull valuation theory, and algebraic geometry.

The basic algebra texts listed in the bibliography

as references [63] - [68] contain more than adequate background material in the appropriate sections.

The final two chapters of this text are, in fact,

somewhat independent of the first six chapters. I recommend that after looking at the introduction, the reader look through Chapters VII and VIII in order to gain motivation for the foundational material of Chapters I through VI.

Introduction

It is my hope that the methods developed in this text will lead to an interesting embedding of algebraic topology in a purely algebraic category, namely, some category of partially ordered rings. At the same time, the theory provides a convenient abstract setting for the theory of real semi-algebraic sets, quite analogous to commutative algebra as a setting for modern algebraic geometry. I might motivate the study of partially ordered rings (somewhat frivolously) as follows.

One observes that the integers, together with

their ordering, is an initial object for a lot of mathematics.

On the

one hand, consideration of order properties leads to the topology of the real line, then to Euclidean spaces, and eventually to abstract continuity and general point set topology.

On the other hand, consideration of

arithmetic properties leads to the abstract theory of rings, fields, ideals, and modules. Following either route, one can go too far.

Completely general

topological spaces and continuous maps are uninteresting. general rings and modules are uninteresting.

Completely

Thus the mainstream in

topology concentrates on nice spaces (for example, polyhedra and manifolds) and the mainstream in algebra concentrates on nice rings (for example, finitely generated rings over fields, subrings of the complex numbers and their homomorphic images.)

The two theories seem to intersect even-

tually in category theory and semi-simplicial homotopy theory. The topologists put back in some algebra and the algebraists put back in some topology.

On the other hand, algebraic geometry works best over

an algebraically closed field and such concepts as manifolds with boundary, homotopy of maps and mapping cones, which are extremely useful to topologists,

1

are not readily available in pure algebra.

A simple observation is that

such concepts are easily described by algebraic equalities and inequalities in real affine space, however.

Certainly all of geometry is deeply rooted

in the study of equalities and inequalities of functions on real affine space.

Even in point set topology real functions and inequalities play a key

role, for example, in the theory of paracompact spaces. The real number field is a formally real, real closed field. In fact, the Artin-Schreier theory of formally real fields is precisely an abstract algebraic treatment of inequalities in field theory.

The real closed fields

are the analogues of algebraically closed fields-they admit no proper algebraic extensions in which inequalities still make sense. It thus seems to me that a true understanding of the relations between algebraic geometry and topology must stem from a deeper understanding of real algebraic geometry, or, actually, semi-algebraic geometry.

Moreover,

real algebraic geometry should not be studied by attempting to extend classical algebraic geometry to non-algebraically closed ground fields, nor by regarding the real field as a field with an added structure of a topology.

Instead, the abstract algebraic treatment of inequalities

originated by Artin and Schreier should be extended from fields to (partially ordered) algebras, with real closed fields replacing the algebraically closed fields as ground fields.

It is obvious that such a

category of partially ordered algebras provides an abstract setting for semi-algebraic geometry (study of sets defined by finitely many real polynomial equalities and inequalities), and it seems plausible that such a category would allow a natural development of algebraic topology and homotopy theory. It is essential that the reader understand that algebraic topology (at least the

homotopy category of finite simplicial complexes and the

study of reasonable functors

on this category) is known to be completely

independent of topology, that is, independent of limits, continuity, the infinite arithmetic of open and closed sets; even the completeness of the real numbers is irrelevant.

The most highly developed reduction of homo-

topy theory to pure algebra is the semi-simplicial, or combinatorial,

approach, developed by D. M. Kan, J. C. Moore, M. M. Postnikov, and others in the 1950's.

The problem with this reduction (ignoring inefficiency) is

that it seems unmotivated without first developing the point set topology of finite simplicial complexes, which, in turn, is founded on the topology of the real line.

Also, differential topology seems unnatural in this setting.

My philosophy is that the derivation of homotopy theory from point set topology is anhistorical accident.

Basically, I regard the real

goal to be a mathematization of our experience^sensation, and perception of space, time, and matter.

This experience is inherently finite, but

involves counting, hence algebra, and order relations, hence inequalities. OUT immediate perception of boundaries of objects, and spatial and temporal order relations justifies a more structured approach than the mathematical reduction of all experience to simply counting small or big finite sets. In fact, it seems to me that a reasonable first approximation of our perception is the set theory of sets in affine space defined by finite collections of algebraic equalities and inequalities, along with the Boolean set theoretic operations of finite unions and intersections, and differences.

This sort of set theory has much in common with topology,

but is fundamentally very different.

Thus we will often use the language

of open sets, closed sets and so on, but it is to be understood that a set in the plane like

y > x

is not open because it is a union (necessarily

infinite) of open balls but rather because it is the set of points where the 2 single algebraic function

y-x

is positive.

In general, functions can be

replaced by their graphs, hence admissible functions from one semi-algebraic set to another can be thought of as certain types of semi-algebraic subsets of the product.

Thus morphisms also avoid the infinite definitions of point

set topology. According to the Hilbert Basis Theorem, any set in affine space over a field defined by algebraic

equalities

by a finite subset of the equations

f

of affine space defined by inequalities f (x,...xn) = 0

f (x^.-.x ) = 0 = 0.

is already defined

Over an ordered field, subsets x

gg( i'-- x n ) :L °

as wel1

as

equalities

are clearly not always representable by finitely many

equalities and inequalities.

It is perhaps this fact which has led to the

divergence of the fields of algebra and topology. Topology is a good example of a subject which produces answers of interest before the real problems are fully clarified.

As several examples

of such answers, which come up in more than one context in mathematics, I would list Lie theory, the theory of compact surfaces, the Bott periodicity theorem and K-theory, the classification of differentiable structures on spheres, the theory of cohomology operations (even cohomology theory itself), the computations of the classical group bordism rings, and the emerging classification of singularities of maps.

These subjects deal with topological

concepts, but also turn out to be related to problems in algebra and number theory.

Thus one feels certain that it is good stuff.

On the other hand, much of geometric topology is concerned with analyzing just what pathology can and cannot occur, using infinite definitions and constructions.

Thus one has space filling curves, but a very strong

regularity theorem about simple closed curves in the plane.

I regard this

as evidence that arbitrary continuous curves tend to be uninteresting, but it is not evidence that simple closed curves are interesting.

In fact, in

three space one has wild embedded arcs and spheres and their classification is not regarded as a mainstream problem.

Under the assumption of topological

local flatness, it is known that every n-sphere in topological ball. smooth n-sphere in n / 3

n+1

space bounds a

More important, however, is the question of whether a space bounds a smooth ball.

n+1

Provocatively, if

the answer is known to be yes, but the proof is harder than the

corresponding topological theorem. smooth copy of

S

in

IR

If

n = 3, it is possible that some

bounds a topological ball which is not diffeo-

morphic or piecewise linearly equivalent to the standard

4 D .

Pathology can involve morphisms, as do these examples, or absolute properties of spaces.

Thus topological manifolds of dimension one and two

are classified, manifolds of dimension three are known to possess unique piecewise linear and differentiable structures, but are not yet classified, while it is still unknown if manifolds of dimension four and higher can always be triangulated.

There is perhaps a widespread feeling that if attention is restricted to, say, differentiable manifolds, pathology disappears.

It is known

(Whitehead) that smooth manifolds admit a unique compatible piecewise linear structure and it is also known (Nash, Tognoli) that compact smooth manifolds are diffeomorphic to non-singular real algebraic varieties.

Also

(Milnor, Serre and many other contributors) any compact manifold admits only finitely many distinct differentiable structures. category allows too many morphisms.

But the differentiable

Any closed set in Euclidean space

can be realized as the zeros of a smooth

(C°°)

function.

The studies of

singularities, diffeomorphisms, flows, and foliations in recent years have produced many pathological phenomena, as well as regularity theorems under suitable hypotheses.

Another area in which there are strong regularity

theorems for the objects is the study of smooth compact Lie group actions on compact manifolds.

The compact Lie groups are algebraic groups and

Palais has extended the Nash-Tognoli theorem to an equivariant version which roughly says all compact Lie group actions on compact manifolds are algebraic, up to isomorphism. My own interest is exactly the reverse of this tradition of seeking regularity theorems in topological situations.

Instead, I advocate beginning

with algebra and working toward geometry, in an attempt to discover just what geometric phenomena are realizable by finite algebraic constructions. It is not so much a question of one type of mathematics being superior to another, but simply a question of how best to understand the dividing line between algebra and topology, and I think this dividing line should be approached from both directions.

The notion of inequalities is very close

to this dividing line, but essentially on the algebraic side.

From the

algebraic point of view it is more or less clear that the real algebraic numbers are just as useful as, or even preferable to, the real numbers. I will return to this philosophy from time to time in the course of this introduction. In any event, in this book, I first develop systematically an abstract theory of partially ordered rings.

The models I have in mind are rings of

real valued algebraic functions on certain semi-algebraic sets in affine

space and their quotients by various allowable ideals.

My axioms thus

reflect properties of these models, and aside from a certain amount of curiosity, the abstract theory interests me only insofar as it contributes to the eventual goals of better understanding semi-algebraic geometry and algebraic topology. The second half of the book is devoted to a systematic introduction to real semi-algebraic geometry via Artin-Schreier Theory and the language of

partially ordered rings.

For the most part, the actual results can be

found in the existing literature.

In particular, the papers by Dubois,

Efroymson, Lang, and Stengle, referred to in the bibliography are very similar in spirit to (and, in fact, influenced greatly) my philosophy. Also, there are excellent introductory accounts of Artin-Schreier theory in the algebra texts of Lang, Jacobson, and van der Waerden. What, then, is a partially ordered ring?

Generally, the definition

found in the literature is a ring, together with a subset of elements called positive such that sums and products of positive elements are positive and such that if

x

and

-x

most efficient terminology is to call positives as strictly positive.) squares are positive. unit.

are positive, then 0

x = 0.

(The

positive and to refer to non-zero

I make the further assumption that all

Also, of course, all rings are commutative with

Given such a set of positives, a partial order relation is defined

on the ring by

x >_ y

if

x- y

is positive.

The definition is purely

algebraic. The assumption that squares are positive is justified, first, because it is true in all the examples I want to fall within the scope of the theory and, secondly, because it seems to be a very useful assumption for proving analogues of the basic results in commutative algebra. It is easy to see that a ring admits such a partial order if and only if the following condition holds:

1 < i < n.

whenever

n 2 Z a. = 0, then each X i=l

2 a. = 0 , X

The set of all finite sums of squares is then an allowable set

of positives, in fact, clearly the smallest such.

Nilpotent elements are

decidedly permitted by this condition. The morphisms between partially ordered rings which are important are

the order preserving ring homomorphisms.

Kernels of such morphisms are

called convex ideals and are characterized by the property that if a sum of positive elements belongs to the ideal, then so does each summand.

This

gives a category, (POR). The category (POR) turns out to be not the best approximation to the ultimate goals.

A useful subcategory is the category (PORNN), partially

ordered rings with no nilpotent elements.

However, nilpotent elements

are actually useful, just as in modern algebraic geometry.

A compromise

is the intermediate category (PORCK), partially ordered rings with convex n killers. The added axiom is the following condition: whenever ( 2 P - ) x = °» i=l 1 p.

positive, then each

p.x = 0 ,

1 1

By semi-field I mean for some

b.

Such

(semi-units) are the analogues of units since they

belong to no proper convex ideal. 8. Certain categorical constructions such as fibre sums, fibre products, direct and inverse limits are carried out in the various categories. 9. The set

X

of prime convex ideals is given the Zariski "topology",

and basic functorial and "topological" properties established.

A

structure sheaf of partially ordered rings is constructed by means of localizations.

The stalks of the structure sheaf are the partially

ordered rings one obtains by localizing with respect to complements of prime convex ideals.

The ring of global sections is generally

larger than the original ring. of

1,

In fact, in (PORCK), the shadow

S(l) = {s _> 1}, consists of non-zero-divisors

and the global

sections of the structure sheaf is the ring one obtains by localizing with respect to

S(l). This is reasonable, since this localization

exactly inverts all elements which belong to no prime convex ideals, that is, "functions nowhere zero on

X".

(In (POR), S(l)

can have

zero divisors, a ring does not even inject into its global sections, in general, and the global sections cannot be described by a simple localization.) 10. A "universal bound" is obtained for roots of a monic polynomial with coefficients in a partially ordered ring. obtained for solutions of of even degree.

More generally, bounds are

f (x) denoted

(We assume

R

real closed.)

x £ Rn

The set

a partial order on 14

'P(V)

V

of polynomials nowhere negative on

RfT-^...^].

g i (x) > 0,

On the geometric side, let

be a Zariski dense set (that is, a polynomial vanishing on everywhere).

with

V C Rn

vanishes V

defines

Artin's solution of Hilbert's 1 7 t n problem can be interpreted as an algebraic relation between the orders

'])

and

£(Rn).

Namely, if

f

is

a nowhere negative polynomial, Artin proved that for suitable polynomials h, h^,

h f = Zh..

In other words, as a rational function, f

is a sum

of squares. Given a partial order to be the subset of those with

p

f C B x G B

not a zero divisor.

generally, define the derived order 'J*, which satisfy

px = q, for some

p,q G 'p

(-p ) = '$ (R n ). w d

Thus, Artin's theorem becomes

The motivation for introducing the derived order is the following. The geometric notion of a function being nowhere negative is, to a large extent, a birational notion, depending only on behavior on a dense open set.

Artin's theorem is quite reasonable, in fact, expected, when considered

from this viewpoint. abstractly yield f

Similarly, an inequality such as

f >_ 0.

However,

f

G'£

does give

f

>_ 0

does not

f G 'p , at least if

is not a zero divisor , and certainly if we seek a function theoretic

interpretation of our partially ordered rings, the deduction of from f >_ 0 is desirable. Consider

$ = ^[

g l

. .. g k ] ,

g±

TR]

{x G R |g-(x) > 0, 1 < i < k}, an open set. theorem reads of all

^ d = "^(U).

U, but decidedly g-(y) 21 °-

Of course,

.

Let

f >^ 0

\}{gl . .. gk>

A generalization of Artin's

^ (U) = $ (U), where

U

is the closure

If may not include certain degenerate points

y, where

Birationally, these degenerate points are lost from the

semi-algebraic set

W = {x G Rn|g.(x) >_ 0}.

Graph of

g

The degenerate points are also lost from the point of view of ideals. The real Nullstellensatz says that the maximal convex ideals of

R[T, ... T ] ,

15

relative to the weak order

in the usual manner of evaluating functions at ideal,

/F

x.

If

I

is any

'p -convex

consists of precisely the functions vanishing on the zeros of

Suppose now

'.p

'£ = ']3 [g-, . . . g-, ].

is replaced by

The maximal

n

ideals are the points

W = {x G R |g.(x) > 0, 1 0, 1 £ i _< k}

as above.

the appropriate strong Nullstellensatz also holds. algebraic.

The set

In both cases, U

is semi-

This is not trivial, but follows from the Tarski-Seidenberg

theorem, which we discuss below. Let order

Q C A = R[T, ... T ]

satisfy

Q = >/Q, and

Q

convex for some

-p C A.

The general theory in this first volume guarantees that the m finitely many associated primes of Q are also 'p-convex, Q = ,H Q..

Also, if

A/Q

is given its residue partial order

convex ideals will correspond to the maximal contain

Q.

'p/Q, the maximal

'])-convex ideals of

However, if one considers other orders such as

A

which ^d^f

CP/Q)^

C P J / Q ) > the maximal convex ideal spectrum may change--certain degenerate zeros of

Q

for

(Remark that even if

'pj.

will not be convex.

For

'JVQ = 0VQ)ci-)

In fact, Q

might not even be convex

'p = '£,, one does not necessarily have

example, when

maximal convex ideal spectrum of

£ = ' ^ ^ . .. g R ] (A, '$) is

Q

in

However, if some

W.

If all

g. £ Q,

"enough zeros" in

Q

are

(A/Q, CP,/Q) )

of

x

all

U

Q

as

is the set

'£, convex. Q

has

Specifically, U

in some neighborhood of an algebraic

'^-convex, the maximal convex ideals of Q

in

U, but the maximal convex ideals

are only those zeros

x

such that every neighborhood

contains an entire neighborhood of simple points.

g A £ Q, then

C£/Q) d = C£d/Q) •

coincides with the order

16

is

"the zeros of

of

in

Q

is prime, the

Q.

Continuing, if (A/Q, P J / Q )

will be

U = {x G R |g i (x) > 0}.

must contain all the zeros of simple zero of

Q

(A/Q, p/Q)

'p, convex unless

n

If, where

Q

W = {x G R |g-(x) >^ 0}

g. £ Q, then will not be

and n

before, and the maximal convex ideal spectrum of of zeros of

I.

If some

g. G Q, obviously

If 'p/Q

'£ [g.|g. £ Q]/Q» so the above discussion applies

with fewer A/Q

g.. In any case, Cp/Q),

consists of exactly the functions in

nowhere negative on the maximal convex ideals of

is a nice generalization of Artin's theorem.

(A/Q, CP/Q) d )•

This

The pictures below indicate

how the sets above can differ.

W ( g i ) , some

gj E Q

SpecflVQ) = * ) 0

P ^/Q)d = )

S ec

0

SpecCJ>d/Q) = •

Z(Q)

Another natural problem generalizing Artin's theorem is to find an algebraic characterization of functions non-negative on the set of all zeros of Q

in W, that is, the maximal convex ideals of

(A/Q, p / Q ) •

This is solved by a theorem of Stengle, which implies that such

f £ A/Q

are precisely those for which an equation

(f 2 n + p)f = q

(*)

holds, with

p, q ۥ$.

(mod Q)

In the language of our structure sheaf of partially

ordered rings on Spec(A/Q, -p/Q), this condition translates to the statement that

f

is "positive" in the partially ordered ring of "sections over the

basic open set D(f)".

Similar results hold for (A/Q, 'P^/Q), characterizing

the functions non-negative on all zeros of Q

in U.

The paragraphs above discuss "irreducible" affine semi-algebraic sets. A general closed, affine semi-algebraic set S

is a union of finitely many

sets, each defined by finitely many polynomial equalities and inequalities. By the affine coordinate ring of S, we mean the ring the polynomial ring by dividing by the ideal on

A(S), obtained from

I(S) of functions which vanish

S, together with the partial order '$(S) consisting of functions nowhere

negative on S. Our general theory allows us to identify convex ideal spectrum of

(A(S), 'P(S)).

S with the maximal

Moreover, two general results allow

us to reduce much of the study of arbitrary

S to the irreducible case.

17

First, the associated primes of any absolutely convex ideal are always convex.

Secondly, if

'£, and

'$„ are two orders on a ring and

'£, 0*.p?, then

prime ideal convex for

Q

is either

(This last result was found by A. Klapper.) two sets

Wj = {x|g i(x) >_ 0, 1 < i < r }

Q

'£, -convex or

is a

'^-convex.

As a corollary, the union of

and

W 2 = {x|h.(x) >_ 0, 1 < j < s}

can be identified with the maximal convex ideal spectrum of the order

Vgi]

nt

ideals

Mhj]'

i£ indeed

I(W,), I(W 2 )

V g i ] ' '^w[hj]

are non-trivial.

are orders

- Otherwise, the

In general, degenerate inequalities

on one variety are handled by passing to non-degenerate inequalities on subvarieties.

Finally, the theorems of Artin and Stengle characterizing

non-negative functions on certain sets can be used to give necessary and sufficient conditions for

f £ A(S)

are purely algebraic formulas for

to belong to

'P(S).

These conditions

f, like (*) above, expressed in terms

of the original finite collections of polynomials which define

S.

Just

as in algebra, where the ideal generated by a set of elements is more important than the specific basis, in our case the invariant notion of a polynomial being non-negative on a semi-algebraic set has more geometric significance than the particular defining equations and inequalities, and yet this invariant geometric notion is algebraically expressible in terms of the defining polynomials. In the end, perhaps the following is the neatest characterization of the affine coordinate rings (A, '.J3).

First, A

(A(S), 'p(S))

among all partially ordered rings

should be a reduced algebra of finite type over

Secondly, there should exist finitely many (0) = n p

and orders

'.jK C A . = A/P^

refinements of the weak order A -^IIA^

w

is the natural inclusion.

ij

are

(A-j/^i)

as

discussed above.

f = A H II1]). ,

The

I

P^

need not be distinct These basic building

The maximal convex ideals will

correspond to the points in the closure of the set of simple zeros Pi

at which all

a formula for

g-.(x) > 0.

(A(S),'^(S))

with

d

nor minimal, although the minimal primes do all occur. blocks

P. C A

which are derived orders of finite

*p. = ('£ [g. .]) , such that •*•

where

'^-convex primes

R.

A set of primes

{P^}

x

of

which leads to such

can be intrinsically described.

We also establish the basic results of dimension theory in our semi-

18

algebraic category.

That is, simple points do exist and the expected relations

between transcendence degree and chains of convex prime ideals hold. real closed field we prove an implicit function theorem. analogous to the classical case of real numbers: algebraic functions defining a germ singular, then

f

f:

if

inverse function theorem.

The statement is

f,,...,f

(R ,0) -*• (R ,0)

has a smooth algebraic inverse near

For any

with

0.

are smooth df(0)

non-

(This is the

Implicit function theorems are routine corollaries.)

Note that if one has an R-valued function germ, it makes sense to ask if it has derivatives.

One should not get carried away and try to study "all

differentiable functions."

There are quite nice relations between formal

algebraic derivations, abstract partially ordered rings, and the usual definitions of derivatives.

e-6

Once the implicit function theorem is available,

it is routine to give Whitney type stratifications of semi-algebraic sets into non-singular manifold-like strata, with any real closed field as ground field. The study of quotients Q

is

'$ absolutely convex

A/Q, where

Q f ft\ is more complicated.

£ = Cp [gi-'-giJ) • W 1 K (J

convexity of all associated primes of of the isolated primary components.

Q

Suppose

General theory yields the

and also the absolute convexity

However, embedded primary components

must be chosen carefully, before one can establish an absolutely convex primary decomposition of

Q.

Very crucial to the argument are (i) the

restriction to finitely generated algebras over fields, (ii) the restriction to absolutely convex ideals (category (PORCK) rather than (POR)), (iii) the specific form of order.

p

as the derived order of a finite extension of the weak

Dropping any of these conditions leads to convex ideals which cannot

be expressed as intersections of convex primary ideals, even for Noetherian ambient rings

B.

A quick example might be in order. isolated prime

(X)

component is also

and embedded prime

(X,Y).

(X ,XY) C R[X,Y], with The isolated primary

(X), and in pure algebra, one has a large choice for the

embedded component, say now, however,

Consider

(X 2 ,Y),

(X2, Y-cX),

(X 2 ,Y 2 ,XY), etc.

that among the order relations are

Any convex ideal containing

Y

primary component belonging to

must contain

X,

0 £ X,

0^ 0.

Again, this branch in the first and third

quadrants has a formal power series description.

In general, suppose

is a maximal convex ideal of some affine coordinate ring assume is an integral domain. the order

'$ at

neighborhood of 'p

m, by taking m

f £ '.p

if

f

(B,'.j>) which we

We can also localize

is non-negative on some

in the maximal convex ideal spectrum of

will be an order if

points.

We can localize, B .

m

m

(B,p).

Here,

is in the closure of suitable algebraic simple

Our criterion is, then, that

m

is a geometric simple point if

the associated graded ring

G(B ) = m

AH(mnBj © n>0 AH(m n + 1 B m )

is a polynomial ring and the positive elements in the induced order are the polynomials which are non-negative in some neighborhood of the origin in the appropriate affine space. respect to the order

'$ C B .

in the order

x

'|3 , so

and

In Figure (b), 0 ,g^ let

is locally non-negative at

It is a theorem that these

allow only non-empty

Rn

x}.

(Thus

V

is

1

the interior of the closure of the set

22

can be interpreted

For each finite collection of polynomials

V{g-i > • • • >g^ IK

positive.

V

R(T ;[ ...T n ).

U V

where all

gi

are strictly

are semi-algebraic sets.)

V, that is, g1 ,...,g,

such that

'$ [g, ...giJ

We is

a

partial order on

RfX^.-X ] .

Then there is a natural, bijective, refinement

preserving correspondence between derived orders on &

in the family of sets

Given a filter provided

&

and filters

'V. Total orders correspond to ultrafilters.

the associated order

f | >_ 0, some

R[T.....Tn]

-pf^)

f E'$( n (-?) = {0} (ii) ? (iii)

+

K ?

a2 G ^

and ?•?• ( B $ g ) , we

f ($.) C ^

We thus have a

category of partially ordered rings which we denote If

(A,$) G (POR),

we refer to

$

as an order on

elements of ty positive and elements of f In fact, the subset a £ b

s

]3 C A

if and only if

(POR).

= p - {0}

A -*- B

notation

v

b - a G |5.

strictly positive. A

by

It is easy to redefine partially

is a morphism in (POR) if

b-a G ^

We call

defines a partial order relation on

ordered rings and morphisms in terms of the relation f:

A.

a £ a'

is somewhat preferable to

B

Trf

a CP a )

C


H

a

^B c B

*p. c A

Obviously, there is such if and only if

$. = ^ , the weakest order on

irf :

A +

B

-

is a ring homomorphism and

One might ask if there is an order

namely, just take

Ki, = 'B... W W M u M I am indebted to Paul Cohen for proofs of these results. In fact, 2 2 h f = g

one can always write x G M.

for suitable

In the other direction, there are

C

f°r

f (x) >_ 0

h,g, if

a

^

nowhere negative functions

on the real line, which vanish only at the origin, but which are not sums of squares of

(5)

C

functions.

We consider the polynomial ring

functions on affine space for all

x G R , then

f

R^.

It is true that if

n = 1

and

is a sum of squares of polynomials.

However, if

Examples are due to Hilbert.

However, if we pass to rational

functions, then every positive polynomial is a sum of squares. Hilbert problem, solved by Artin. If

f G R[X. .. . X ] in

then for suitable R[X, . . . X ] order

and

^

.

and

f (x, .. . x ) > 0 in —

h, g i G R[X X . .. X J ,

h ± 0,

(6)

42

in

h2f =

2 g 2.

If ^

is the weak

is the order defined by positive functions on

2 h f =

x = (x1 ... x ) G R ^ , then x

(x, ... x ) G R ( n ) , in

for all

is an integral domain, of course.

c *£ , . since if

borhood of

This was a

We may rewrite the result as follows.

then Artin's result becomes in our notation D£

f(x) _> 0

f(X, X 2 ...X ) >_ 0, which are not sums

n >^ 2, there are polynomials of squares.

R[X, ... X ] , ordered as real

R^

n

.

^

(

. C D^^ .

R^

,

But also,

m

2 S g.

h

Since

and if

f(x) < 0

for some

vanishes identically on some open neighh

is a polynomial, this implies

h = 0.

Return to the general construction of Example (2), (A,^) G (POR),

g S

is a set.

Suppose

BC A

is a subring.

separates points of S, by identifying all

We may as well assume

s and s

1

B

if f(s) = f(s') G A,

f e B. Suppose

Y C S

is a subset such that

f G B, f | Y = 0 implies

f = 0. We call such a Y • B-Zariski dense. the weakest topology on S

for which all sets

Y

is, in fact, dense in Z = {s g(s) = 0} are

closed, g G B. A if

B-Zariski dense

gIY — °

Y

as a f u n c t i o n

(

determines an order on B, say P Y , by Y -• A ) .

Thus ^ g

gef

Y

is the affine order on B,

which we defined earlier. Clearly, if Y, and Y_ are two such sets, then implies $ cp . Y C Y 1 l Y Y 2 l More generally, let ^ be a family of B-Zariski dense subsets of S. Suppose and

&_ 1,

Otherwise,

for some

q i G Q,

p^ G $,

i = 1,2, with 0 < a 2 r < qx + p x b 2

0 < a 2 s < q2 + p 2 c 2 .

Multiplying these equations gives

0 < a2(X+S) < q € Q ,

which is a contradiction.

Corollary 2.3.7.

Let

I C A

be a convex ideal.

n P

Proof.

Then

p.

convex prime I C p

We apply 2.3.6 to the partially ordered ring

(A/I,

and use the proposition of the next section.

51

2.4.

Relation Between Convex Ideals in Fix

(A/p) E (POR), and let

I C A

(A,?)

and

(A/I, !J)/I)

be a convex ideal.

Proposition 2.4.1. (a)

There is a natural, bijective, inclusion preserving correspondence

between ^-convex ideals J/I

of

J

of

A

which contain

I

and

p/I-convex ideals

A/I.

(b)

For any such

J D I, there is a natural isomorphism in (POR),

(A/J, $/J) 3- (A/l/j/I,

(c)

The prime convex ideals of

A

which contain

bijectively to the prime convex ideals of ideals of

A

which contain

convex ideals of

Proof.

and

J/I

Given a ^-convex

is

J C A/I

'p/I-convex. is

correspond

and the maximal convex

correspond bijectively to the maximal

A/I.

order preserving morphism Thus

I

A/I

I

p/I

J D I, the identity on

A

induces an

(A/1, *p/I) •> (A/J, !p/J), with kernel Conversely, if

convex, then

IT: A -> A/I

J = TT'^J) C A

is

J/I.

is projection, ^-convex.

The

remaining details are equally simple.

2.5.

D

Absolutely Convex Ideals Although the category (PORCK) is a full subcategory of (POR), one

sees fewer ideals as kernels of morphisms in (PORCK). of the category (PORNN), where obviously an ideal

I c A,

is the kernel of a (PORNN)-morphism if and only if "P-convex.

The same is true (A,$) E

I = /f

and

(PORNN),

I

is

In this section we will investigate kernels of morphisms in

(PORCK) . Let convex if

(A;p) E (POR), 0 ^ SL ^_ b

absolutely convex if

52

and

I C A

an ideal.

bx E I

(p+q)x E I,

implies

We say that a x e I.

p,q E*p, implies

I

is absolutely

Equivalently, px E I

and

I

is

qx E I.

Proposition 2.5.1. (a) Absolutely convex ideals are convex. (b) Arbitrary intersections of absolutely convex ideals are absolutely convex. (c) Any convex ideal

I with

I = /f

is absolutely convex.

In

particular, convex prime and maximal ideals are absolutely convex. (d)

If y c $ »

is a refinement of order and

convex, then (e)

If

I

is £'-absolutely

is ^-absolutely convex.

(A,?) -> (A 1,^ 1 )

f:

I c A

absolutely convex, then

is a morphism in (POR) and I = f" (If) c A

I ! C A1

is

is absolutely convex.

Proof. (a)

Let

x = 1, the unit in

A.

(b) Obvious. (c)

If

0 ± a (PORNN), defined by assigning to partially ordered quotients

(A,?) G (POR)

(A/AH(0), ?/AH(0)) G (PORCK)

and

the

(A//o, ^//o) G

(PORNN). Proof.

2.6.

Whatever remains to be proved will be left as an exercise.

•

Semi-Noetherian Rings It should not be surprising that various finiteness conditions on partially

ordered rings lead to structure theorems and results which cannot possibly be proved in general.

Since a partially ordered ring consists of a ring

A,

together with an order '£ C A, natural finiteness conditions can involve either the '^-convex ideal structure or the order 'p itself, as an extension of the weak order *pw c A.

We are ultimately interested in partially ordered

structures on finitely generated extensions of real closed fields. Chapters 7-8.)

(See

Such rings are, of course, Noetherian in the classical

sense, hence no finiteness assumptions are necessary on chains of convex ideals.

However, even for this restricted class of rings, many classical

results fail to generalize to the partially ordered context.

For example,

one cannot always decompose convex ideals in such a ring as intersections

56

of primary convex ideals. the order.

One needs additional finiteness conditions on

When one examines the classical proofs of primary decomposition

and other results which fail to generalize the point which causes the difficulty often involves the fact that in commutative algebra the multiples of a single element always form an ideal, while in partially ordered algebra the smallest convex ideal containing a given element may be quite large. On the other hand, chain conditions on convex ideals have certain interesting consequences, regardless if the underlying ring in the classical sense or if the order

'p

A

is Noetherian

satisfies extra conditions.

Some-

times these results are proved most easily in the classical context of Noetherian rings as applications of primary decomposition, rather than by exploiting the chain conditions directly.

Thus it actually gives one some

added insight into these classical results to investigate chain conditions in the partially ordered context, where certain techniques of proof are unavailable. It is the purpose of this section to initiate this study.

We work

entirely in the category (PORCK) because of our frequent use of the quotient construction

(I : X)

for ideals

I C A

and subsets

X C A.

However, 2.6.1 through 2.6.4 have obvious (POR) versions as well.

Definition 2.6.1.

A ring

(A,'£) e (PORCK)

is semi-floetherian if any

of the three equivalent conditions below hold: (i)

The absolutely convex ideals of

(A,'p)

satisfy the ascending

chain condition. (ii)

Any non-empty collection of absolutely convex ideals of

(A,}5)

contains a maximal element with respect to inclusion. (iii)

Every absolutely convex ideal I = AH(x1,...,x, )

I

of

(A,'.J5) may be written

for some choice of finitely many elements

x r . . . , x k e I.

As simple applications of the definition, we state some standard results.

Proposition 2.6.2.

If

(A,'.j>) e (PORCK)

is semi-Noetherian,

I C A

57

A 1 = A/I, then

is absolutely convex and any refinement

Proof.

'£' of

(A','-P!)

p/I.

This is immediate from 2.5.5 (b).

Let us call an absolutely convex ideal I.

is semi-Noetherian for

absolutely convex, implies

Proposition 2.6.3.

If

I = I.,

•

I

or

irreducible if

1 = 1 ^

i

I = I~«

(A,'£) G (PORCK)

is semi-Noetherian, then

every absolutely convex ideal is a finite intersection of irreducible absolutely convex ideals.

Proof.

An absolutely convex ideal maximal among those not so

expressible leads to an immediate contradiction.

Proposition 2.6.4. I = /F

(A/p) e (PORCK)

is semi-Noetherian and

is a radical convex ideal, then there is a unique expression

I = P.n...n p 1 K if

If

i ^ j.

where the

P. j

Moreover, the

containing Proof.

P.

are prime convex ideals and

are precisely the minimal prime ideals

In fact, one only needs the ascending chain condition for The proof is just as in the classical case.

Uniqueness is clear since if a prime ideal

P

P

contains some

D P! D P .

P^

for some

also shows that the taining

P. £ P. 1 J

I, which are necessarily convex.

radical convex ideals.

then

•

Thus if

i,j , hence

P.

P

contains

P^.-np^,

P ^ - . - n p ^ = p|n...np^, j = 1

and

Pj = P!.

then

The argument

which occur are exactly the minimal primes con-

I, which are thus necessarily convex.

(This result will be

generalized in section 3.9.) Existence of the stated decomposition is established as follows.

Each

radical convex ideal is a finite intersection of irreducible radical convex ideals, by the chain condition on radical convex ideals. is an irreducible radical convex ideal, then ab € I, let

a,b £ I.

{P^,}

containing

58

Let

{P }

is prime.

I = /F

Otherwise, let

be all the prime convex ideals containing

containing

a

and let

be those

Pa

b.

I = n P by 2.3.7. a oi

Then

I

But if

{PaiJ

be

those

I,

Pa

On the other hand, {P }= {P ,}U{P „} a a a

since

I = I1 n I"

ab G I, hence 1

a e I ,

I1 = n p , , I" = n P „ .

where

b e I", this contradicts the irreducibility of

Since

I.

•

Directly from 2.6.3, absolutely convex ideals in a semi-Noetherian ring can be understood in terms of irreducible absolutely convex ideals. However, irreducible absolutely convex ideals need not be primary, even if the ring is Noetherian.

Despite the lack of primary decomposition,

the associated primes of an absolutely convex ideal are accessible in any semi-Noetherian ring.

Proposition 2.6.5. I C A

Suppose

(A/p) G (PORCK)

an absolutely convex ideal.

is semi-Noetherian,

Then the following sets of primes

coincide: (i)

{P|P

prime and

P = (I : x ) , some

(ii)

{P|P

prime and

(I : x)

(iii)

{P|P

prime and

P = /I : x, some

(iv)

{P|P

a minimal prime containing

is P-primary, some

Passing to

x G A}

x G A} (I : x ) , some

Moreover, this set of associated -primes of

Proof.

x e A}

I

x G A}.

is non-empty and finite.

(A/I, 'p/I), we may assume

I = (0). Certainly,

Set (i) c Set (ii) c Set (iii) c Set (iv). Moreover, among all the absolutely (0 : y ) ,

convex ideals (0 : z) = P

is prime.

y i 0, choose a maximal one, say For suppose

abz = 0 ,

hence by maximality, (0 : z) = (0 : bz) Next, suppose ideals if

(0 : y)

c £ P,

P G Set (iv), say

with

(0 : z) = (0 : cz) C P.

bz ^ 0.

(0 : z) = P.

abz = 0 ,

Namely, if

This proves

Then

In fact, if

bz ^ 0, then

(0 : z) C (o : b z ) ,

(0 : x ) .

(0 : z)

Thus

(0 : bz) C P,

c £ P, hence

(0 : z)

Among the

be maximal.

bz t 0, then

We now claim

Then

Thus Set (i) i 0.

minimal over

Otherwise, bcz = 0, some

= (0 : z ) , contradicting

az = 0.

P

az = 0.

(0 : x) C (0 : y) C P, let

(0 : z) = (0 : bz).

hence

and

bz ^ 0.

(0 : z).

b G (0 : cz)

is prime, hence

(0 : z) = (0 : bz), hence

Set (iv) c Set (i).

Finally, we must prove there are only finitely many associated primes. Suppose the set of associated primes is there are infinitely many.

Let

{P } = {(0 : x )}

AH({xrv}) = AH(x

••• x

).

and suppose If

x,, / xrt ,

59

then

P

2.5.3.

^ P

K P = ( O : x ) D H ( O : x ) =

, but

Thus

Pa D Pa

f

i

, some

i.

3,

0

3^33

many

P Y

all

j , then

infinite chain

P

C Po

Proposition 2.6.6. I,J C A

PD D Po P ^ Pi

properly contain some C P

I.

Then

P.DP . P t %

some

PQ . Po

Suppose

J c P., some

Let

P.,

Conversely, if

xJ C I,

ideal of the form prime of

x £ I, then

(I : J) ^ I.

In

consists of all elements

x.J c I, hence

(I : J) t I.

j

J C (I : x)

is contained in some maximal

I.

•

If

(A/p) e (PORCK)

I.

P

Proof. Let

is semi-Noetherian and

containing

I

I, and

Suppose

(I : x) £ P, where

xy G I,

y and

orders on

!p-2

C S, as in Example (6) of 1.7.

n $

S, with values

That is, for

j = 1,2,

s € S.}.

is clearly the affine order corresponding to the subset

s e S.

functions

'£ 12 = ^

'#., '$„ could be affine orders defined by suitable

? 1 2 = {f e A | f(s) >. 0, all

Points

Then

could be a ring of functions on a set

•j). = {f 6 A | f(s) >_ 0, all

Then

A.

$u.

in an ordered ring, and S.^ S

two

We are interested in relations between the convex ideals

For example, A

subsets

c A

define

f G A

with

S.US^S,

s e SjUS^.

'p.-convex ideals, j = 1,2, namely, the ideal of f(s) = 0.

Thus points of

S1US2

define

£12

convex ideals. More generally, for any

A

and

•J^-convex or ^-convex, certainly j = 1,2, then

I. n I

I

'.p, and

'^ , if

is '^12-convex.

is '^12-convex.

I c A If

I.

is either is '^.-convex,

Both these remarks are equally obvious

for absolutely convex ideals. If

I C A

is

'^12

below that we can write convex ideal, j = 1,2.

convex and

I = /F

I = l^ n I , where

I. = /T.

is a .p.-radical

For non-radical ideals, even primary ideals, such

a decomposition is impossible in general.

62

is radical, then we will show

We will illustrate these phenomena

with examples in 2.8. The main positive result in this direction is the following very useful fact, discovered by Andrew Klapper.

Proposition 2.7.1 (Klapper). ideal, p 1 2 = ^

Proof.

n

Then

'P 2 «

Suppose

P

p

is

Suppose either

was neither

P C A

is a prime 'pi:?-convex

'^-convex or

'p.-convex nor i

0 £ x £ y (rel 'pj), 0 £ u £ v (rel 'P2), with

'^-convex.

'p0-convex. £

y,v e p,

x,u

0 < x 2 u 2 < y 2 u 2 < y 2 u 2 + x 2v 2

(rel $.)

0 £ x 2 u 2 £ x 2v 2 £ y 2 u 2 + x 2 v 2

(rel 'P ) .

£ P.

Choose

Then

and

Thus

0 $ 1 2 = 'j^ n f2

orders

'.p, 2-convex ideal [resp. primary P = v^

be the (prime) radical.

i-s '^-.-convex [resp.

1.-

Then if

"£, absolutely convex].

Proof. Since

P

Suppose

0 <x

2

Assume

x u z E Q

as

Choose

in the

and

If and

u,v

'.p,) and

as above.

proof of 2.7.1, and

2

x z E Q.

as is easily checked. finally

0 K

of

K.

(ii) The (POR)-automorphism group of % ( A r ) group

S

which acts on

A

exactly to the inclusions of r-simplices in If

K, L

is exactly the symmetry

by permuting vertices.

(iii) The surjective (POR)-morphisms

(iv)

correspond

A ^ K ) •* ^(A 1 *)

correspond

K.

are two finite complexes,

We thus obtain a faithful embedding of the category of finite simplicial complexes in the dual category of (POR).

(11)

In this example we study the ring of polynomials

rational numbers.

It is convenient to think of

say on the real line. think of

Q[X]

Q[X]

Q[X], Q

the

as a ring of functions,

For algebraic purposes it is slightly preferable to

as a ring of functions on

Q, the real algebraic numbers.

Before beginning the discussion, we indicate two directions of generalization.

First, Q

real closure dense in

F.

could be replaced by any totally ordered field The discussion below will apply to

F, in the interval topology.

examples, no element of

70

F

F, with

F[X], provided

(This is a tricky point.

F

is

In all

will be infinitesimally small relative to

F.

That is, all intervals around F.

0

in

F

will contain non-zero elements of

However, there are examples where some intervals of

elements of

F.)

F

contain no

Secondly, the number of variables could be increased.

This would require much more work.

In fact, the study of polynomial rings

in several variables over an ordered field is, in some sense, the final goal of the whole theory.

Polynomials in one variable is essentially

zero-dimensional algebraic geometry, which is trivial from a geometric point of view. All ideals in convex ideals rings

Q[X]

We will (i) characterize the

(f(X)), and (ii) characterize certain orders on the quotient

Q[X]/(f(X)),

be given.

are principal.

f i 0.

Since this is only an example, proofs will not

We fix the weak order ty C Q[X]

is known that

throughout this example.

It

$ w , the sums of squares, coincides with the affine order,

the polynomials which assume no negative values on the real line.

Assertion 1.

An ideal

irreducible factor of Assertion 2.

Suppose h.

(f(X)) C Q[X]

f(X) has real roots.

All convex ideals

(f(X))

is convex if and only if every

is convex.

(f(X)) C Q[X]

Factor

f, say

are absolutely convex.

k i f = h. 1

k r h^ , where the

are distinct irreducible polynomials, with real roots.

real roots of

h.

by

a-••

We denote the

By the Chinese Remainder Theorem, we have an

isomorphism of rings

i=1

(hj1)

From 1.7, Example (11), we know that any order on a direct product of rings is a direct product of orders. h-

Let

Thus we are reduced to studying

k i Q[X]/(h. ) ,

irreducible. g = g Q + g 1 h i + ••• + g ^ ,

If

g(X) e Q[X], write

j,

0 k.

We have (whence

where

degree (g.)< degree

g. i 0.

g e ? w /(h i i ) C Q[X]/(h i i ), the weak order, if and g = 0) or

j < ki

is even and

g i (a i -) > 0

for all

71

real roots

a.,

of

h..

Assertion 3 can be interpreted geometrically. zero set of

h^.

Regard

neighborhoods of to

Z..

k.-th order on

Q[X]

Z. = {a--}

be the

as a ring of germs of functions, defined on

The ideal

Z..

Let

k i (h. )

is the ideal of germs which vanish g £ (h.1)

Assertion 3 says that a function

is

k

i Q[X]/(h. )

positive in the weak order on is locally non-negative near

if and only if the germ of

g

Z..

The geometric interpretation of the weak order suggests other orders on

k i Q[X]/(h i ) .

denote by

Each root

ou . , aT. .

1>S C Q[X]/(h i i ).

ou . has two sides on the real line, which we

Any non-empty subset

A function

g £ (hu1)

S C {a.. , aT.}

belongs to

$g

is locally non-negative on those sides of those roots the subset

k i The orders P s C Q[X]/(hi )

Q[X]. The maximal orders on

by single elements of h^. y>s

if the germ of a..

g

picked out by

S c {at. , aT.}.

Assertion 4. on

defines an order

k i Q[X]/(h. )

are all quotients of orders are precisely the orders defined

{a-. , aT.}, that is, by one side of a single root of

The maximal orders on

k i Q[X]/(h. )

are thus total orders.

All the orders

are (PORCK)-orders.

(12)

If we refine the order on

convex ideal changes. algebraic numbers.

For example, let

Y

is

*PY

in Example (11), the notion of

Y C Q

be any infinite subset of

(Infinite is equivalent to Zariski dense.)

1.7, Example (6), define Let

Q[X]

_ 0, all

Then an ideal

convex if and only if every irreducible factor of

one real root in (13)

As in

h(X)

y € Y.

(h(X)) c Q[X] has at least

Y.

In several examples above we have studied polynomial rings with

orders defined in terms of values of functions on subsets of affine space. There is a more intrinsic, algebraic approach to these same orders. Let with C] I

(A/P) e (POR)

{l a }

= (0). Then the natural map

define a refinement of

72

and let

s

£

s

be a collection of ^-convex ideals, A -»• IT A/I

by ]5 = A Q ^1 Q)/I ) . a a

is injective, and we can

We are in the situation of

1.7, Example (2), where

A

is now interpreted as a "ring of functions" on

the index set {a}, with value at a The order ^ C A (i) All I

in the partially ordered ring

(A/I ,ty/I) .

can also be characterized as follows,

are ^-convex

(ii) ?/I a =F/I a CA/I a (iii) f

is the union of all orders on A

satisfying (i) and (ii).

As a specific example, let A = H[X,«*'X ] , with the weak order ^ . Each

y €= IR> ' defines an evaluation homomorphism

a maximal *$ -convex ideal.

(In fact, the I , y e Ec ^, are exactly the

maximal ^ -convex ideals of A.) Certainly, quotients

on R

(n)

f] f % I = (0), and the

y € R are all isomorphic to R, which has a unique order. The

A/I

order ty C A

A -*• IR, with kernel I

is thus the affine order, of polynomials nowhere negative

. (2X,X 2 )CZ[x]

(14) Reconsider the convex but not absolutely convex ideal of Example (5). The associated primes are is not convex.

(X) and

(2,X).

However, (2,X)

This illustrates an advantage of the category (PORCK), where

associated primes are always convex. (15) Consider R[X,Y]

and the family of orders p n = '£W[X,Y, 1-Y, Y n -X],

We have P n c $ n + 1 since

n^l.

(X2, XY)

The ideal 2

is !p -absolutely convex for all n. To see this, 2

write

(X , XY) = (X) n (X , XY, Y m ) .

and

2

m

(X , XY, Y )

Y n - X = Y n + 1 - X + Yn (l-Y) . Let p = U ^ .

It is relatively easy to check that

are $ -absolutely convex if m > n.

Thus

(X)

2

(X , XY) is

'^-absolutely convex, since a contradiction of this assertion would involve only finitely many elements, hence would already be a contradiction in some :p . The associated primes of observed that

(X , XY)

(X2, XY)

are

prime

(X,Y).

We have just

has an absolutely convex primary decomposition for

each of the orders '$> . However, (X , XY) for

(X) and

has no convex primary decomposition

$> since, first, the primary component corresponding to the minimal (X) is necessarily

(X) itself in any decomposition, and secondly,

73

any

'£-convex ideal with radical

0 < X < Y

m

(rel

(X,Y)

Y m , hence

contains some

'£) . 2 (X , XY)

The ideal

is actually irreducible among

'^-convex ideals. If

but

(X^, XY) i I

since

X, since

then clearly

X £ I- n l o . 1 z

^

/ i ^ = (X),

= (X). We must then have

It follows that

(X) C / T ^ ,

/I~ must be a finite intersection 2

(X, Y - c i ) , corresponding to points on the Y-axis.

of maximal convex ideals

(X2, XY) = Ij n i 2

But now Proposition 2.6.8(c), (e) would imply that

has a

primary convex decomposition, which we know is impossible by the above discussion.

(16)

(Andrew Klapper)

'$. C A, neither

j = 1,2, prime £ ..-convex nor

We give an example of a ring

"p.-convex ideals

'p2-convex.

Note

P., such that

A, two orders I = Px n P 2

is

I = /F , so Proposition 2.7.1

does not extend to radical ideals. A = R[X] x IR[Y].

Let Then

'J^ n $ 2 = ^

P 2 = (X) x H [ Y ] . (but not P,

n

?2

^

= iyX-1] x ^ ,

x -j^, the weak order on Then

P2

is

Let

similarly, P x n P 2

Geometrically, the ring

A

n P2

E(E-l) = 0,

is

$ -convex But

would imply

'^2-convex.

can be rewritten as a quotient of a

polynomial ring in four indeterminates, 1R[X,Y,E,F]. by the relations

Px

'^,-convex).

(X,Y) ^ p

is not

x ^[Y-l].

?x = H[X] x (Y),

.p -convex (but not

'^-.-convex since otherwise

(1,Y) e P 1 n p 2 .

A.

£2 = ^

?x n P 2 = (x) x (Y). Clearly

'^-convex) and is not

Let

F(F-l) = 0,

Namely, one divides

E + F = 1,

XF = YE = 0.

The

maximal convex ideals for the weak order then correspond to the points on the two lines in F = 1,

X = 0.

IR

defined by

The variable

parametrizes the second. P x = (E,X,Y),

(17) ideals convex,

Q X ,Q 2 Q2

F=0,

Y = 0

and

E = 0,

parametrizes the first line, while

In this notation

'^ = '^[X-E],

Y

£ 2 = £ w [Y-F],

P 2 = (F,X,Y).

We give an example of two orders

'^,-convex nor

74

X

E = 1,

with is

^

= v ^ = (X,Y)

such that

^-absolutely convex, but

.p2-convex.

Since

Q

'p., '.p2 on

Q

=

Qi

Qx n

is primary with

IR[X,Y], primary is

Q2

is

'^-absolutely ne

itner

T/Q = (X,Y), we see

that Proposition 2.7.1 does not extend to primary ideals. •p1 = ? W [ X , Y , Y 2 - X ] ,

We let 2

Q 2 = (X ,Y). convex. if

* 2 = '^[X,Y,X 2 -Y].

It is not difficult to check that

In fact, Q x

is

Let

Q1

is

^-absolutely convex, where

0 £ x,y,y -x}. Similarly,

•^ = {£ e H[X,Y] |f (x,y) ^ 0

is

Q2

2

),

'^-absolutely

^

= {f eH[X,Y] |f (x,y) >_ 0

'p'-absolutely convex, where

0 £ x,y,x 2 -y}.

if

Q^CX^

Q 1 n Q 2 = (X 2,Y2,XY)

But

2 is not

'PT-convex since

0 < X < Y

(rel

'$,) and is not

'po-convex since

2

0 £ Y £ X (rel -P2). (18) a

The example in (17) still leaves open the hope that if

'^ n ' ^

convex primary ideal in some ring

'^-convex and

'^-convex, then

where

£.-convex.

Q.

is

Q

A

with

P = v^Q

Q

is

both

Q = Q-^ n Q 2

might always be written

However, we now give an example where such a

decomposition is impossible. Let Clearly

A=H[X,Y], Q

is not

£ -convex since In fact, let

^

= $ W [Y,X-Y],

'^-convex since

0 £ Y £ X (rel

Q

is

Suppose

Then

'P(S)-convex.

(In fact, Q

0 £ f £ g

as functions on

(3f/3y)(0,0) = 0.

and

is

v = i + j

S

and

vectors along the

x

v

S, and both

w

lie in

and

y

and

Q

is not

*^(S) = and we will

g e (X,Y ) = Q.

Equiva-

f(0,0) = 0

and

£(0,0) = 0. f

and

g

g

i, j

axes, respectively. f

and

'£.. n $ -convex.

We must show

w = -i + j , where

and

Q=(X,Y2).

'^(S)-absolutely convex.)

(8g/9y)(O,O) = 0.

It is obvious that

Let

'^ n $ 2 c -p(S)

We consider directional derivatives of

and

is

S = {(x,y) e IR2 | 0 £ y, y 2 £ x 2 } , and let

lently, g(0,0) = 0

the vectors

'p^

'P 2 ). However, Q

0 £ Y £ -X (rel

{f e H[X,Y] |f (x,y) >_ 0, (x,y) e S}. show

? 2 = ? W [Y,-X-Y].

with respect to

are the usual unit Since

vanish at

0 £ f £ g

on

S,

(0,0), we must have

0 (AT,?T)

( commutes. (d) !PT = (i^) ^ , the weakest order on A~ such that

i T is order

preserving.

Proof.

We first show that our definition

a/t E $ T

does not depend

on the choice of representative, a/t, of the element of A,p. at't" = a'tt"

then

for some

t" E T.

We thus have, for any

If a/t ~ a'/f s E T,

Cat)(t')2 (t") 2 s2 = (a l t')t 2 (t") 2 s 2 .

ats 2 E ? , then

Thus, if (a)

a ' t ' ( s t t " ) 2 € $ , hence

a f /t' E ?

r

The equations in A

(alt2 + V l

5 1

^

s

l S2

=

show that if a ^ t j , a 2 t 2 E ? T , then Also, since

(a/t)

2

2

= a /t

2

2

Vl

S

l *2

a^t^

and a t

2

2

+

¥ 2 S2 * !

&2/t2, (a^tj) (a 2 /t 2 ) E ? 2

= a t -l

2

r

E $, we see that P T

contains all squares. Finally, suppose

a/t E ^

T

and -a/t E !PT.

ats 2 s 2 £ 0 £ ats 2 s 2 , hence

Then

^T C N 1

Say, ats 2 £ 0 £ ats 2 .

at(s s + ) 2 = 0 and a/t = O G i ^ .

Thus

is an or< er

^*

(b) is trivial. (c)

We already know that there is a unique such morphism

in the category of rings. check that

78

g

Namely, g(a/t) = f(a)f(t)~

is order preserving.

1

1

E A .

g: A™ -* A' We need to

Lemma 3.1.2. x,y

invertible.

(A 1 ,? 1 ) G (POR), and let

Let

Hence

A ! , with

in

0 ± y"1 ± x" 1.

Then

a/t G ^L,, then for some

The lemma implies our result, since if a t s 2e p .

0 < x f y

f (a)f (t)f (s) 2 G $'

s G T,

f (a)f (t)f (s)2(f (t)"1£ (s)" 1 ) 2 =

and

g(a/t) The lemma itself is easy, since

3.2.

y~

= y(y~ )

y

- x~

Sufficiency of Positive Multiplicative Sets In general, if

T c A

other than elements of

T

is a multiplicative set, there are many elements which are invertible in

T = (a G A|ab G T, some

then if

and

T

is exactly the set of elements of

ab G T, then

then

1

b/ab = a.' G A T

axs = ts G T

for some

A .

b G A> ,

A

invertible in

and, conversely, if

s G T.

In fact, if

Note that

T

A^.

First,

(a/1) (x/t) = 1/1 G A T , is a multiplicative

set.

Proposition 3.2.1. TCA

as above.

If

T

f

Let

(A/p) G (POR),

T C A

a multiplicative set

C A

is any multiplicative set with

T C T' C f,

there is a natural isomorphism in the category (POR)

defined by i T ^ r ( a / t ) = a/t.

Proof. clear that

It is routine that i T T , CPT) c ? T i -

morphism in (POR).

course.)

is a ring isomorphism.

It is not quite so clear that

However, since elements of

the universal property of -* (A T ,^ T )

iT T,

(AT,^T,)

is a morphism in (POR).

T

1

It is also

i~ T ,

is a

are invertible in

in (POR) implies that

i^ T ,:

AT,

(A^,^,

(This could also be proved directly, of E

79

Turning the argument around, we see that there are multiplicative sets smaller than

T with the same localization in (POR).

Proposition 3.2.2.

Let T C A be a multiplicative set

(A/P) e (POR).

Define T + = {t t G T n T 2 = {t2|t e T}. T c T+ c T

Then

are multiplicative sets and the natural maps

(A i 2, ? T 2 ) ~* (AT+, ? T +) ~* (AT, ^ T ) i li 11 are isomorphisms in (POR).

Proof.

The first assertion is trivial and the second follows from

Proposition 3.2.1 and the observation that

T C T + C T C (T 5 ).

D

Thus, when localizing in (POR), we can get by with positive multiplicative sets.

3.3.

Refinements of an Order Induced by Certain Localizations Suppose

i T : A -> A T

is injective, (A,!p) G (POR).

Then there is a

natural refinement of *£, namely, the contraction of *PT to A, that is, ^ c i*(iT)^p = f T n A = {a e A|as 2 G f, some

It is easy to see that *$„ n A = *p + n A = !B 2 IT I 3.2.2 of the preceding section. then

T

+

If A

= ^ , the strictly positive elements of A.

This is exactly the derived set Dv£

80

A, say from Proposition

is an integral domain and T = A - {0},

^T+ H A = {aGAlpaG'P

domains.

n

s G T} .

of $

some

In this case

p G "P+ } .

discussed in 1.6 for integral

More generally, if

(A,^) G (POR)

is arbitrary and

plicative set of non-zero divisors of We conclude that any order where

T C A

iT:

coincides with

$

is the multi-

A -• k^, is injective.

has a natural refinement

is the set of non-zero divisors.

der-ived order of

*p, = ^

We will call

n A,

^,

the

(hopefully remembering that the derived set

D^

*p , only in special cases).

Proposition 3.3.1.

Proof.

^

A, then

T

If

If

T C A

(A,$) € (POR), then

CPd)

= 1^.

is the set of non-zero divisors, i^:

A -• A T

the

canonical map, one has (as a special case of a more general principle) that (iT) i*(iT) $ = (iT) p C A T . i

*

l

i

*

i *

Remark.

Now intersect with

Note that if

A

A. Q.

of

D

is an integral domain

the following "simpler" definition of an order fractions

A.

i

A

(A,*P) G (POR), then

^/-Q^ °n the field of

does not work:

The condition doesn't always say the same thing for the equal elements and

as/bs

2.4.

of

A, . unless the order

Convex Ideals in The morphism

enables to

(A,P)

iT:

and.

^

is sufficiently strong, say

$ = ^

(A^ ? T )

(A,p) -> (AT, !f T ),

T C A

a multiplicative set,

us to define correspondences between ideals of

I C A, assign the ideal

a/b

I T = IA T C A^, and to

A

and

A™.

Namely,

J C A T , assign the ideal

i^(J)-C A. If *P

(A/P) G (POR)

convex, since

i^

and

J c AT

is

$T

is a (POR)-morphism.

convex, then

i^.1^) C A

is

The other correspondence also

preserves convex ideals. Proposition 3.4.1.

If

I c A

is ^-convex, then

IT C AT

Moreover, there is a natural isomorphism in (POR), induced by

is iT:

PT A

81

where

IT:

(A/p) -* (A/I, y>/I)

Proof.

One knows from commutative algebra that

a ring isomorphism

(A/I) ,„,. 25, A^/IA

with forming quotients." to be

is the quotient projection.

(Namely,

.

iT:

A -»• A T

induces

That is, "localization commutes

Acp/IA^

has the right "universal property"

(A/I) (T>jO Consider the commutative diagram

A/I

Following the arrows clockwise from be extended, first to order

$

that =

A, we see that the order

$/I C A/I, then to

c (A/I)^..

can also be extended going counterclockwise from

*PT C A,j, can be extended under is

IT

flVl).^

^ C A

A.

can

Thus the

It follows

TF, which can only happen if kernel (IT)

1$T- convex.

The equality of orders weakest extensions of

Remark.

CP/I)^,-^ = ? T / I T

follows since both are the

$ C A, under either i • TT or

The result that

1^

is

i ° iT»

*£„,-convex could also be proved by

direct computation.

Proposition 3.4.2. (a) spondence If

J

If

J C AT

J ^ i

is an ideal, then

(J)

is injective from ideals of

is a primary ideal of (b)

If

Q C A

disjoint from

Q «• QA T T

A ^ then

i^. (J)

is a primary ideal and

a primary ideal, v ^ H T = 0, correspondence

J = i~ (J)AT.

Thus the corre-

AT

to ideals of

is a primary ideal of

Q n T = 0, then

/

v Q A T = /QA T , and

AT>

A.

QAq, C A ?

1

Q = i~ (QA T ).

is

Thus the

is a bijection between all primary ideals of

and all primary ideals of

A.

A

The correspondence commutes

with the nil-radical operation and preserves inclusions, hence also provides a bijective correspondence between prime and maximal ideals of

82

A^. and

prime and maximal ideals of I C A, then

A, disjoint from

T.

(If

I n T t 0, any ideal

IA ? = A r )

Proof.

This is a result in commutative algebra and doesn't really

involve (POR).

Of course, we get a result in (POR) as a corollary by

replacing the word "ideal" in the proposition by "convex ideal."

This

is justified by Proposition 3.4.1 above.

3.5.

•

Concave Multiplicative Sets (A/p) e (POR).

Let p 6 S

implies

is convex.

q e S.

A subset

S C A

equivalently, S

is ooncave if

is concave if and only if

Y C A, we denote by

S(Y)

Thus, S(Y) = f l S a

containing

Y.

S S Q (Y)

Given any subset

the smallest concave multiplicative set containing

where the

S

a

run over all concave multiplicative sets a

We call

We construct

Then

A- S

Arbitrary unions and intersections of concave sets are concave.

We are interested in concave multiplicative sets.

Y.

0 £ p £ q,

S(Y)

S(Y)

the concave shadow (or, simply, shadow) of

explicitly as follows.

Y.

Let

o°°

is the smallest multiplicative set containing

is the empty product.)

Y.

(1 G S Q (Y)

Let

S n + 1 (Y) = S Q (S n (Y) U { q | 0 < p < q, p € S J Y ) } ) .

S m (Y)

Then each

is a multiplicative set, S m (Y) C S

is the concave shadow of

Proposition 3.5.1. Let in

(A/P) -> (A1 ,$•)

f: A

and

morphism

t E T, then g:

(

A

S(Ty

(Y)

and

S(Y) = U S n ( Y )

Y.

Let

T C A

be a multiplicative set

(A/£) e (POR).

be a morphism in (POR) such that wherever f(s)

is invertible in A?

^s(T)^ "* (

^')

such that

A1. the

0£t£s

Then there is a unique dia

gram

83

commutes.

Proof.

First we prove a Lemma.

(S(T)) 2 C S(T 2 ) C S(T + ) C S(T).

Lemma 3.5.2.

Proof.

(

m (T))

in

(S Q (T)) 2 = T 2 = S Q ( T 2 ) .

S Q (T) = T, so

and s

Only the first inclusion is not obvious.

2

2

C Sm(T ).

S m (T)

(Sm+1(T))

S

Any element of

2

C Sm+1(T ).

Thus

(S(T))

2

S(T) = U S (T) n n

Suppose, inductively,

m+1(T)

is a product of elements either

or larger than positive elements of 2

But

S m (T).

It follows that

2

C S(T ).

D

Returning to the proposition, from 3.2 we have canonical isomorphisms in (POR) (we suppress the symbols for the orders)

A

We now replace suitable

g:

( A S m' ^SfTp

S(T 2 ) ^ A S(T + ) ^ A S(T) '

by

^AS(T+V ^S(T+ )^

(AS(T+^> ^SfT+'P "*" (A* $'^ '

But

Siven

a n d look for a a

positive multiplicative

+

T ,

set

S(T + ) = {s|o £ t £ s, t € T + } ,

as one verifies easily by showing the set on the right to be a concave multiplicative set.

Thus, the hypothesis of the proposition clearly gives

( A s ( T + ) , ? S ( T + ) ) •+ (A',P f ), as desired.

g:

3.6.

The Shadow of Let

1

(A/p) G (POR).

set, we study

As an important example of a concave multiplicative

S(l) = {b € A|l £ b } , the shadow of

First, an element

84

D

y G A

is invertible in

1.

Acr .

if and only if

1 .:

A -• A c r i .

oil)

need not be injective.

Z>{i.)

In fact, kernel (i c r i .) = {a G Alab = 0, some

If

1 < b, then —

0 < a —

implies

a

= 0.

kernel(ig.,.), which one sees by

2

(a.^ + a 2 )

pathological.

G

a1,a2

2 expanding

a G kernel(i c r ,O ^HlJ

< a b , so —

Consequently, 2a1a2 = 0 for any

b, 1 < b} .

and

^

- a2) .

It is the ideal

K

We regard this kernel as somewhat

of

2.8, Example (4).

We now give an alternate characterization of semi-fieIds. Proposition 3.6.1. if and only if

igriy

such a case, Ag,..

Proof. A

Let A

(A/p) G (POR).

"*"Asfl1

is

in

Then

3ective

is the field of fractions

Semi-fields were defined in 2.3.

is an integral domain, so

paragraph above.

i,.,...:

and

A. .

If

i s a fielc

^sfl") of

^#

In

A.

(A,$)

is a semi-field,

Moreover, directly from the definition of semi-field and A

are invertible in

Thus, A S ( 1 ) = A ( o ) , a field.

Conversely, if

igQ-j = A ^ A g , .

then all non-zero elements of

A

is injective and

are invertible in

paragraph above, this says exactly that

We point out that any convex proper S(l).

is a semi-field

A -*• A« .

•

Our next observation is that if set, then the construction of

(A/J5) G (POR),

(A^/p^) E (POR)

S C A

a multiplicative

in 3.1 is elementary, as are

the results in 3.4 concerning the correspondence between prime

'£c-convex o

ideals of

S, and the

Ag

and prime

f-convex ideals of

A

disjoint from

permutability of localization and residue ring constructions. just proved, we know there exist prime

'pg-convex ideals of

enables us to shorten certain arguments. Proposition 3.9.2.

Let

A g , and this

For example, we can reprove 2.^.1.

(A,'£) G (POR),

I C A

a convex ideal.

/T = n p , the intersection taken over all prime convex ideals

90

By the result

P

Then

D I.

Proof.

Passing to

(A/I, *-p/I), we may assume

trivial step is showing that if ideal

P,

f £ P.

ideal of {f }

n,

f

£ 0

all

Consider the localization

A-^ corresponds to a prime

I = (0). The non-

n, then for some prime convex (A,,, 'P~).

'^-convex ideal of

Any prime A

'p--convex

disjoint from

as desired.

•

The use of localization also aids in the study of minimal convex primes.

If

(A,^) e (POR)

of multiplicative sets S(l), the shadow of any such prime

S

S C A

I

is a convex ideal, consider the family

with

S n I = 0.

1, are such sets.

is contained in a maximal

'$„ -convex ideal containing

P , with Thus

and

I C P Q C A- SQ

S Q C A - PQ

and

Since our original

S .

Consider

gives a prime

IA g

(A - P Q ) = 0.

could have been

have proved the following

(POR)

and

Zorn's lemma easily implies that

(such primes do exist since

in

S

For example, {1}

(Ag , '£s ) . Any 'p-convex ideal A g /IA g

By maximality of A-P

t 0).

SQ,

for any prime

S Q = A - PQ.

P 3 I, we

extension of a familiar result in com-

mutative algebra.

Proposition 3.9.3.

If

(A,'.p) e (POR)

ideal, then the minimal prime ideals of 'p-convex.

3.10.

Any prime ideal containing

A I

and

I C A

containing

is a I

£-convex

are always

contains a minimal such prime.

Applications of Localization, II - Zero Divisors Our next application of localization will be to zero divisors.

The

results are most satisfactory if attention is restricted to the category (PORCK). If

A

is any ring, let

and let

N

denote the non-zero divisors.

algebra is that

D

D

denote the set of zero divisors of

A

A standard result in commutative

is a union of prime ideals.

More precisely,

D

is the

union of those prime ideals which are minimal over ideals of the form (0 : a ) , a e A.

Since

(0:1) = (0), the minimal primes of

The proof of these assertions is as follows. union of the ideals

A

are contained in

Certainly

(0 : a ) , so if we prove that whenever

P

D

is the

is a prime

91

D.

minimal over

(0 : a ) , then

P C D, we are finished.

maximal multiplicative set disjoint from x £ P-D

(0 : a).

But If

S = A- P

is a

P £ D, choose

and observe that

(J S x is a multiplicative set strictly n>0 S, but disjoint from (0 : a ) . This contradiction proves P C D.

larger than

Using the results that if and that if

(A,'£) e (POR), any minimal prime is convex

(A,'p) G (PORCK), all ideals

minimal primes over

(0: a)

are convex, hence so are

(0 : a ) , we see that we have proved the following.

Proposition 3.10.1. (a) then

If

(A,'p) G (POR)

and

P

is a minimal convex prime ideal,

P C D. (b)

If

(A,'p) e (PORCK), then

ideals minimal over ideals

D

(0 : a ) ,

is the union of all convex prime

a £ A.

•

For the sake of argument, we give two alternate proofs that in (PORCK),

D

is a union of convex prime ideals.

These arguments are

analogous to standard arguments for commutative rings. First, if fact, if

0 0 1 I = H ( P ( n V m ) ) , then I(P) = P ( n + m ) .

If

There is a completely analogous construction for absolutely convex ideals, using absolute hulls, A H ( I 1 + I 2 ) ,

AH(I 1 I 2 ), etc.

As final applications of localization, we discuss the semi-Noetherian case.

If

(A,'£) E

(PORCK)

any multiplicative set

is semi-Noetherian, then so is

Suppose

is a multiplicative set, I C A Then the associated primes of P C A

Proof.

(A,'£) E (PORCK)

I = (0). If

I Ac

in

Ac

are exactly those primes

I

disjoint from

P = (0 : x) C A

is prime and disjoint from

PA g = (0 : x)A g = (0 : x/1) C A g .

prime of

in

Ag.

P = AH(y1...y,)> with

Conversely, suppose

(A/p)

P

exactly the ideal

PA C

S.

I, then the ideal {x E A|xy E I

some

S, then it is

PA g

is an associated

PA g = (0 : x/s) C A g . s. E S.

is an associated prime of

is semi-Noetherian, I C A

associated prime of

Thus

xy.s. = 0 E A, for suitable

(0 : xs-...s,) = P C A, hence

If

S.

(A g/IAg , '-Pg/IAg) = ((A/I) g , CP/I) S ), we may assume

easy to check (0)

is semi-Noetherian, S C A

an absolutely convex ideal disjoint from

is an associated prime of

Since

for

S c A, by 3.4.2(a).

Proposition 3.11.2.

where

(Ac, $ c )

Let

Then (0)

in

absolutely convex, P C A

A.

an

I (P) = kernel (A -* A-.p./IA,p.) y £ P}

considered in 2.6.

•

is

According

to 2.6.8(e), I = n I(P.), the intersection taken over the finitely many associated primes of

I.

by 3.11.2 and 2.6.8,

IA g = n IAgCPJVg), the intersection taken over those

associated primes and

Pi

If now

of

I

S C A

disjoint from

I(P-) = kernel (A -• A C /I(P.)A C ). 1

O

is a multiplicative set, we have

1

S.

Easily, IAgCPJVg) = I(P i )A g

Thus we have proved the following.

O

95

Proposition 3.11.3.

If

absolutely convex, S C A

(A/£) E (PORCK)

is semi-Noetherian, I C A

a multiplicative set, then

kernel (A -• A^/IAg) =

n I(P.), the intersection taken over those associated primes disjoint from

Remark. of

I

of

of

I

S.

•

As a special case of 3.11.3, let

and choose

I

P.

f. E A

not contained in

with P.

only associated primes of

f. E P. - P

Let I

f = ITf.

disjoint from

P

be an associated prime

for each associated prime n

and let S

S = {f }

_.

P.

Then the

are those contained in

P.

(J (I : f*) • Since (I : f) c (I : f 2 ) C . . n>0 is an increasing chain of absolutely convex ideals, we must have I(P) = I(P) = kernel (A -> A /IA f ) =

Thus

1

(I : f n ) , for suitably large

n.

In the classical Noetherian case the ideals the intersection of the primary components of of associated primes contained in of an -isolated set

E

P.

I

I(P)

belonging to the set

is a union of

Ep.

of associated primes, meaning

E

contains all asso-

The most general such

Thus for any isolated set of associated primes

the intersection of all primary components of E

Ep

This is a special case of the notion

ciated primes smaller than any one of its elements. E

are interpreted as

I

E,

belonging to primes of

is invariant and is characterized as

Q I(P)« In our semi-Noetherian PGE case we do not have primary decomposition in general, but the results above and in 2.6 show how many of the classical results involving isolated sets of associated primes do extend.

3.12.

Operators on the Set of Orders on a Ring Let

to

(A,£) £ (POR).

'$ a refined order

potent operators

We are interested in operators

Aj>.

( A $ = Ap)

A

which assign

In particular, functorial operators and idemare natural objects of study.

The motivation is ultimately this:

We seek to interpret partially

ordered rings as "rings of functions" in some generalized sense. formulas involving a "function"

96

f £ A

ought to imply

f

Certain

positive.

Thus

if

f

is positive and

a set, then

f

so should be '.p W

(A,'.p)

is a ring of suitably valued functions on

should be positive.

f.

But if the order

If 'p

p

and

(l + p)f

are positive,

is too small, say a weak order

or a finitely generated refinement

'p [g-,*#*gv], such function theoretic W 1 K

results will not follow by simple algebraic manipulation. natural algebraic extensions of

Thus we seek

'p, which more closely capture the behavior

of functions. We present four such operators here.

The first three are related to

localization. The operator

^.

If

(A/P) e (POR), (PORCK), or (PORNN), and

NCA

is the multiplicative set of non-zero divisors , then we have seen that iN:

(A,'p) -* (A N /P N)

'£d = A n !pN p

of

'p.

is injective and induces a natural refinement Specifically, f G '£

not a zero divisor.

The operator shadow of

$ .

(A,'p)

Let

p,q G 'p,

originally.

The operator

'£, is idempotent,

(A,'£) G (PORCK) or (PORNN).

Then

S(l), the

1, is a multiplicative set, contained in the non-zero divisors. iaf,^'b(.lj

We thus have an injection

fs

pf = q, for some

(A/p ) e (POR), (PORCK), or (PORNN),

We have

whichever category held

if

t h e r e f i n e m e n t of

'£

g i v e n by

(A,'£) -• (A c r i ., f c n j , and we denote by oil) oil)

Anf

Thus

f e f

if

( l + p ) f = q,

e

P,q ' P We have

(A,$ ) G (PORCK) or (PORNN), accordingly.

the absolutely convex ideals of verified that

operator

'pg

Given

'£ S

(A,'£)

and

(A/p )

is idempotent, 'P = CP ) • S S c

is a functorial operator in

In 3.8 we saw that

are identical.

Certainly

fc c 'p,. S Q

AN)

f = q/p,

p,q E'p,

From the point of view of functions, this gives information about p.

On the other hand, if

gives a globally defined formula for the function 'p

The

(PORCK) or (PORNN).

f E '£,, we can write formally (or in

least off the zero set of

We also

f.

f G 'pg, then

f, at f = -j^-

Thus functions in

are "non-negative" in rather a strong way. Suppose

with

A

is a ring, and

n p^ = (0). Then

A

Pi C A

is a finite collection of primes

has no nilpotent elements and

A

has only

97

finitely many minimal primes, say Moreover,

Hp

= (0)

and

Up

P , which are included among the

P..

is exactly the set of zero divisors of

A.

Proposition 3.12.1. In the situation of the paragraph above, assume either an order

'£ C A

with all

Pi

or assume orders

'.p. C A^

A -> II A^.

$ d = A H nCP a ) •

Then

Proof:

and set

convex, and set

'•]) = A n n tp^.

There are two points here.

then 'J5

p/P

Pi

than

and

£ C A

will be strong enough to agree with 'p, C IICP )

not a zero divisor,

suppose given

(A/P ,'£/P ) •

f £ A

a

G

P , since

h f = q (mod P ) , where

and since E h

a

tp C A H n 1^i. £ PQ p

A .

Still

h

pf = q,

p £ P .

£ P

p,q *='$,

Conversely,

and q (mod P )£ '£

Adding these equations gives

for any minimal

(A^'-P^,

n p - p , we may assume ijta 1 A, hz f = q £'£, since a a

n p. - p , and we may assume equality in i a and

Secondly, if

on

is clear since an equation

i^a

(0) = H p

'£

can

A n nCP ) •

can be reduced modulo

with

$,

is defined using all the

Multiplying each such equation by a square in h

(A,'p), then

will generally be a much weaker order than

The inclusion p

Pa

Consider the inclusion

First given

be computed by passing to the integral domains there are more

(A^'^) = (A/P^'p/P

P o , we conclude P

(Eh^)f = E q a G 'p

f £$,. a

•

The geometric significance of 3.12.1 will come out in Chapter V M , when we relate derived orders with positivity conditions on certain "non-degenerate" subsets of real varieties.

The non-minimal primes

correspond to proper subvarieties (hence lower dimensional or "degenerate") of the irreducible components of a variety.

The operator

$ .

Let

(A/p) G (PORNN)

canonical injection where the Consider those elements p,q G'.p. order

Such

f

{P }

and let

A -> n A/P

be the

are the minimal convex primes of

f G A, such that for some

A.

n

(f + p ) f = q,

n > 0,

form a set closed under products and contained in the

A H ITCP/P ) .

It is not clear under what conditions this set will

be closed under sums. In any event, define

n^^ >^ 0,

98

p.,q i G'p}.

Then

'£

!p

= {£ G A|f = E f ^

is an order and

2n. (fi + Pi) f i

:p C rp .

=

If

^i*

A

£or

some

has only

finitely many minimal primes U p

P , then the zero divisors

This can be used to show

The operator

$

Rewrite

is functorial on

(f

+p)f = q

positivity condition on the shadow of

'J>d = A H IICP/Pa)

f

as

f = q/(f

n

+ p) .

off the zero set of

(f n + p)f = q £ |

i(f) E *-P «(£"»•

are exactly

In this case

f.

In fact, if

S(f)

e

(A/-P) -+ ( cff) »'

can be seen to be equivalent to A^-*

ought to be a ring

f.

Q> hence each summand

these summands are in on

k, all

'^.

is

A

It is not difficult to see that any 'p-convex prime ideal Q C A k 2m +Vi)£± = P ^ i e ?• T h e n '^-convex. Suppose X f± € Q, {f± k k 2nd S ( n (£. + p O J f x J j=l i=i 1

'$ c'$

This provides a strong

i = ig/-£_0.

n _> 0}.

This is

essentially a theorem of Stengle [22], and generalizes in several directions the work of Artin on representation of positive functions [4 ] . It is easy to check that all and that

*p m

is an idempotent operator

Finally, suppose relating the operators

(a) CP»

'^-maximal convex ideals are

n

T)

CPm) m

=

f • m

'p!, "p" are two orders on *p , 'p , p ,

'P -convex

A.

We state some formulas

above and intersections.

s

(b) CPf n ' P M ) p

('£' n ' P M ) d

The inclusions (a), (b) are trivial, as is If

f e T

p f ,p M

d

n f » ,

let

p'f = q',

not zero divisors.

Then

p"f = q",

pSq'Gf,

c

'Pd n '^ d •

p",q" € J>»,

(pfp") 2f = p'q'Cp") 2 = p"q M (P') 2 G 'P' n '^

Thus f'd n •pjj c fpi n -p") d . It is not quite as easy to formulate a result for the operator since even if the Jacobson radicals of clear when the Jacobson radical of

1

'p

'£' n $"

and

tp" vanish, it is not

vanishes.

However, in the

algebro-geometrical situations to be studied later, our rings will have the property that for each maximal convex ideal a fixed real closed field

'p ,

(A/p)

Q,- A/Q

R, therefore with a unique order.

is

In the

reduced case, (A/p) G (PORNN), the Jacobson radicals will vanish and (A,'P )

identifies with a ring of R-valued functions (ordered by func-

tional values) on the set of maximal convex ideals of such orders

'£','£" C A

the maximal convex ideals of

exactly the union of the maximal convex ideals of and clearly

100

$'

n

'P") m = '£'m n '£"• m

'P'

(A,p). p

1

n 'p"

and

For two will be

'p!f by 2.7.2,

IV • Some categorical notions

4.1.

Fibre Products Proposition 4.1.1.

The categories (POR), (PORCK), (PORNN) admit

arbitrary fibre products.

Proof.

The construction is identical in all three categories.

We

first show that direct products exist. Let fa:

(Aa, p a ) G (POR).

(C,^c) -* (Aa, $ a )

a unique (POR)-morphism diagram commutes for all

Consider

(A,?) = (II A a , II? a ).

If

is a family of morphisms in (POR), then there is f:

(C,^c) -• (A/p)

such that the following *

a:

(A,

Thus, (II A a , n f a )

is the direct product of the family

More generally, suppose

ga:

morphisms over the fixed base (Aa, $ a )

over

(B, *PB)

contraction of the order

(Aa, P a ) -> (B, *Pg) (

(B, ]5fi) .

is the subring II^ .

(Aa, $ a ) . is a family of

Then the fibre product of the n A a c IlAa, together with the

B That is,

II

(A , y )

is the ring

together with the order

n A ® B C

and

B -* A ® B C

A ® B C are

In general, however, this cannot be done.

Instead, we construct the smallest ideal

R C A ® B C

which satisfies the

following condition: k 2 .2 (Pi ® q±)xl G R, i= 1

Pi

G '?A , q. G P R , x. G A ® B C

4.2.2. implies

(p. ® q-)*- G R,

1 < j £ k.

This is exactly the generalized extension condition of 1.3 for the pair of maps

A-»AB, C

is then the ring

B -• A ® B.

The fibre sum

(A/BJ

c

A

® (B,^D) B (C,?c)

(A ® B)/R, together with the order C

k

? = {^2

? (p. (8) q.)x*) |p. G ? A , q. G ^

x. G A ® B}

1— 1

where

L

IT: A ® B -• (A ® B)/R C C We verify that

is the projection.

(A ® B/R, !p) C

Suppose given (POR)-morphisms

f:

has the desired universal property. (A,!PA) -»> (D,?D)

such that the diagram below commutes.

102

in (POR)

and

g:

(B,?fi) -* (D,

Then there is a unique ring homomorphism' f ® g: A ® B-*D such that f is A-^A® B-*D and g c

c

k

? B -* A ® B -> D. Let I = kernel (f ® g). Suppose 2 (p. ® q.)x7 G I, x C i=l * p. G ( P A , q i e ? B , Xj_ G A ® B. Since f ® g(( P j ® q.)x2.) = f (p..)g(q..) (f ® g(x. ) ) 2 e ^ D is

1 < j < k, we deduce

(p. ® q.)x. e I, 1 < j < k. Thus,

~ "~ 3 3 3 — — factors A ® B -» (A ® B)/R -» D. Clearly, (A ® B)/R •* D C C C hence

((A ® B)/R, $ ) = ( A ® ? . ) A

c

®

R C I, and f ® g

is order preserving,

(B;f R ).

(c;pc)

In the category (PORNN), the ideal

R c A ® B must satisfy R = v^R, C There is a smallest such R, and the rest of the proof is

as well as 4.2.2. unchanged.

In the category (PORCK), the condition 4.2.2 is replaced by

R, p i e f /

q.

1=1

G p B > x. ,y e A ® B,

1

1

1

,

,

4.2.3. implies

(p. ® q.)x?ye R,

Again, there is a smallest such

1 < j < k.

R, and the rest of the proof is the

same.

•

4.3.

Direct and Inverse Limits Let

I

be a directed, partially ordered set, with relation

{(A., ? - ) } - F T

be a family of partially ordered rings.

Proposition 4.3.1. k < Jj —

in



e

i £

e

i »

Z(I i ) t 0,

i = 1,2, then

1 G H(I 1 + I 2)

By 2.2.4, we can write

Spec(A/p) = Z{e^) U Z(e 2 ),

i = 1,2.

Conversely, if Spec(A,'p) = Z(I X ) U Z(I 2 ),

elements.

i'e2

i = 1,2.

Proof. Z(e x )

G

Z(I 1 )

and

n

Z(I 2 ) = 0, and

I1 n I2

1 £ ej + e 2 ,

consists of nilpotent

e^ G I i n p .

(ej e p n = 0,

If

we get 2n-1 2n-l n n < (ej + ej) = a^e') + a 2 (ej) , 1 = lZn

with

a i G!p,

hence

i = 1,2.

1 ^ H(ei),

If

e i = a i ( e p n G!p.

Set

i = 1,2.

Also, e ^

1 £ e1 + e 2

1 £ H(ep,

and

A -> Ag.,.

and an order

$

S

Q^

C

Recall Acfli

Spec(A,'^) = S p e c ( A s ( 1 ) / p s ( 1 ) ) .

Proposition 5.5.2.

1'A2C

A

in

(A,^) G (PORCK), then

e . , e . e f C A,

Ag,.y

Spec(A;p) for

suitable

is disconnected subrings

i

In 5.5.1 we proved that if

are elements Now,

A

S(1)'

Proof.

If

( A S Q ) » ^ S ( 1 ) ^ ~ ( A i^i) x (A2'^2^

if and only if A

e| e I.,

(A/p) G (PORCK), we can push this result somewhat further.

that by 3.8.4 we have an injection with

= 0,

Since

u

Spec(A,$)

1 £ e, + e 2 = u,

is invertible and

is disconnected, there

e^ £ e^ ,

e.e2 = 0,

1 ^ H(e i ).

1 = e,/u + e2/u = ej + e'.

Consider the natural projection

The kernel since gives

110

of

TT is

AH(eJ) n AH(e^).

e!e* = 0, we have

But if

e!x' = elx1 = 0.

(0: e!) = (0: AH(e!)).

Thus

x

1

xf G AH(eJ) n AH(e^), then,

This follows from 2.5.3, which

= (ej + e^)x! = 0, and

IT is injective.

If

b ! ,c' £ A S . 1 . , then

7T(b'e^ + c'ep

Thus,

cr = c f e ^ + c l e ^ , and hence

b 1 = b'e^ + b f e^,

= (b',c») e Ag (1) /AH(eJ) x Ag (1) /AH(e») .

IT is surjective. b',c' e f p s ( 1 ) , then

Next, if Thus,

TT~

b'e' + c' e[ e ? s ( 1 ) , since

is order preserving and

Finally, 1 £ H(e i ) C A

TT is an isomorphism in (PORCK).

implies

1 £ AH(e i ) c A, since

Then, also, 1 £ AH(ej) C A g ^ . , since A n AH(e i )A s . 1 . = AH(e i ).

e!G

Ai^e^

AH(e£) = AH(ei)Ag

Thus the factors

= /AH(e7)

and

Ag^./AHCep

are nonzero.

D

5.6. The Structure Sheaf, I - A First Approximation on Basic Open Sets We will eventually impose on rings.

The classical idea is this:

a E A, we know "a hence

Spec(A/p)

a ^ 0 e A/Q.

is never zero".

on a basic open set That is, if

Q e D(a).

set

{al}i>0-

a

Specifically, let If

D(a) C Spec(A,*P),

Q G D ( a ) , then

So we localize and invert the element

But in our partially ordered ring if

a sheaf of partially ordered

A

S(a)

over D(a).

we know more than just

a £ Q

be the shadow of the multiplicative

b G S(a), Q e D(a), then

is positive, since then

a

a £ Q,

b £ Q.

$Cr * O D(^a j

(A cr \>'#Qr O >

we mav

as w

^ll assume

a

is positive, (see 3.2 for details).

Thus it is natural to begin by trying to invert all elements of over the basic open set

D(a). That is, to the basic open set

we assign the localized (POR) this assigns

(Ag^^g^^)

Lemma 5.6.1.

If

(A c . .,'£c, O £>laJ ^laJ to Spec (A,?).

D C a ^ = D(a 2 ),

a

^

b^a j

-,,a2

G

Since

S(a)

D(a) C Spec(A,'p),

Spec(A,'^) = D(l),

A, then there is a canonical

isomorphism in (POR),

Proof.

DCa^^) = D(a 2 )

primes containing

a«

implies that the primes containing

coincide.

Thus

y/H(a,) = Ai(a^).

Since

a^^ a

and the a

i> o

e

P»

we have by 2.2.3

111

0 £ a^1 0

since now all elements of

S(a 9 )

and

p,q e:p.

The proposition follows

are invertible in

In fact, this argument really shows that if

" = *D(a2);D(ai):

defined by so that

and vice versa.

r

DCa^) C D(a 2 ), then

0 < a"^1 < pa 2 , and there is a canonical morphism in — * —

a, G ,/H(ao), hence l z (POR),

A

(A

r

(Pa?)

^.a i

S(a 2 )'*S(a 2 ) ) -* ( A S(a 1 )'*S(a l ) ) - *"

r

e Ag,

1#

. , where

a € A, b >^ a^

These morphisms

a ^ ,

are our

candidates for the presheaf restriction maps corresponding to the inclusions D(a,) C D(a 2 ).

of open sets

Lemma 5.6.2.

If

D(a Q ) C

D(a 1 ) C D(a 2 ), then the diagram below

commutes

where

Proof.

This is easy using the universal property of the localizations

(A cr ^y^cr 0 o^a^j b^a.j

If

Q c A

(rather than the explicit formulae for

is a convex prime ideal,

the concave multiplicative set There is thus a natural map

112

A - Q

a ^ Q, then

contains



^^ a S(aa])

From 5. 7.1 (a), the map

$cr on

. C Acr .

l

Secondly, D(a.) C D(a)

the element Since

x/b

means that

and integers

r. > 0.

.) n A C , ..

That is,

= x

Moreover, since

p >^ 1,

U l. , we assume •H J

*D(a),D(a.) ( x / b )

D(a) =

K

p = 1 + P j + ••• + p k G

Then

is compact, hence there is a finite with

a , say, a,,...,av

p. Gtp

a.

( A g ( a ) f ? s ( a ) ) - H ( A g ^ ,» s(gfl) )

. = (npc,

fcr

G •£.

is the contraction of the natural product

First, D(a)

a

for some

all

a

IlA c r v a S(a a )

Proof 5.7.1.(a). subset of the

D(a) = U D(a ) , a,

(x/b) = 0 e A

We claim moreover that

the natural order order

Let

b > a s , some integer ~~

P S / b P S ' Cbps > a s p s > a ^ S , hence a s p s >_ a s , hence

bp s >^ a s

s > 0.

Then

bp s e S(a.).) and we can write

113

x/b = xp s /bp s e A s ( a ) .

To assert that

means that there exists a

*D(a)^

c. > a. 1 , with

r

0 < a

integer s

x/b = xp /bp

s

But now,

= 0 G Acr . o ^aj

Proof 5.7.l(b). above.

i -

a

for suitable

X X

xps c = 0, and since

q. G •£, and some *

c G S(a), we have proved

as desired,

The proof is a slight modification of the argument

xp S /bp s >_ 0

i1

with

in all

Ag,

.,

1 £ i £ k, then there exist elements

in

A.

If

xbp2s(cp2^0

xbp S (c') ^ 0

0 £ q! , we see that But

q.c. = c

aGv / H({a 1 1 ,.. . ,a. k } ) , Ik

implies

We continue with the same notation.

If C

J

D(a) = U D(a.) x i=l

k t. k < 2 q.a. < 2 X x ~ i=l ~ i=l

r > 0.

11

F

l—i

hence

(xp s/bps ) = 0 e A ^ ^

xp s c. = 0 G A.

l—i

We now use the fact that

^

in

0 < a r ' _ 0

c' G S(a), so we conclude

in

2 q ! ( c ! ) 2 = c'

>_ 0

A g . ..

D

The "second sheaf axiom" for basic open sets would say this: D(a) = U D(a ) ot o^

and suppose

x / b G Acr . ot ot, o (_a I

fx /b ) = ip

if> (Note

Ot p

such that

Ot

DCa

, for all

Then there is an element

ct, 8

x/b G A c r . o (_B-J

p

} (x/b)

Suppose

have the property that

(x /b ) G A

D(a a o ) = D(a ) n D(aD).)

in A.

= x a /b a G A s ( a ^

for all a.

This assertion does not seem to be true in general.

However, the failure

of the second sheaf axiom does not really affect the way one defines the structure sheaf.

What it does affect is the evaluation of the global sections.

We will return to this point in the next section. What we can show is that in the category (PORCK), the second sheaf axiom for basic open sets is true.

Lemma 5.7.2. a G iB, and let x /b

a a S AS(a )

Let

(A/p) G (PORCK), D(a) G Spec(A,'f)

D(a) = U D(a ) a a have the

Specifically

be a cover,

114

Suppose the elements

Pr°P«rtX * hat *D(a ))D(a a . ) a

W),D(aaJ(VVeAS(aaq) (X p

G •£.

a

a p

a

01 p

a basic open set,

f o r a11 a

C

W

=

a p

> 6 - Then there exists

x/b6Ag(a) with » D ( a ) > D ( a a ) W W = * A 6 A S ( a a )

f

° r a 1 1 "•

Proof.

The proof is essentially the usual proof for commutative rings. k First, D(a) is compact, so D(a) = U D(a.), for some finite subset of the

a .

If

x/b G X

and

x/b

restricts to

x./b.,

^laJ

CX

1

have the same images in all

A~,

1 < i < k, then

x /b

,

1

..

But

01 Ot

D(a)= U

D(aa.),

and we already know from the "first sheaf axiom", Lemma 5.7.1, that an element in

A

. *-acr

x/b

Thus

is uniquely determined by its restrictions to the

r

must restrict to

x /b , all

a.

Acr

.. ctai

This reduces our problem to

the case of a finite cover. From the hypothesis, there are elements such that

x i - b . - c i . - x . * b i - c i . = 0, all r

0 £ ( a ^ . ) 1 ^ £ c^.

c^. G A,

i,j. Since

(A,$) G (PORCK),

r

x^b. ( a . a . ) - x . b i ( a . a . ) = 0. Replacing x i /b- by x ^ . / b ^ T , we may assume x.b. = x.b.. W em a y also assume that 0 < as . < b., l < i < k. 1331 — 1 — 1 — — Since the Pi G £

D(a i )

cover

D(a), a G /H{a ••-a,} , and we can find positives

such that

t 0

In

A , . b(^aj

1 a•

we have

1 = b/b =

k Pi 2 (-=--)b. .

^_^

D

1

Consider the element

\c

If

This proves that

vn. . n .

.(x/b) = x./b. G A c r

.,

x/b = V

1 < j < k, as desired.

5.8. The Structure Sheaf, IH - Definition Let

X = Spec(A,'p)

with the Zariski topology.

€L, of partially ordered rings on If

U C x

X.

is open, consider the product

jects naturally to

U C X. A section

s:

II ( A r m ,'£r m ) , which pro-

QGU W j U ->• n A QGU

x

n/bn

G

ArnV

and elements

We will define a sheaf

w

Q G U , is continuous if there exists a cover x /b G A c r ., such that for all ot ot o^a 1

WJ

say

s(Q) =

U = U D(a )

Q G D(a ) , (r(V,^x),'j3(V,^x)) induced by the obvious projection

s: U -• n A.o. , s(Q) = xo/bo G A r o v ^ ^ ^J Q€U ^ J

Proof. Suppose

is continuous

»

say with respect to a cover U = U D(a ), and elements x /b e Acr .. a a a a ^*-aa^ Cover V, V = U D(a'). Since V C U, we have V = U D(a a'). Then the P a, 3 a ^ 3 sL : V -• n A. . is defined locally by the elements v W QGV

section

^D(a ) , D ( a a ' ) ( W (X

d

*^U V ^

= s

eA

S(aaL)'

ThuS

S

I V ^ continuous and we define

CX p

p

lv G ^W,0y)>

^y y

Proposition 5.8.3. The ^x),!p(V,^))

is obviously order preserving.

(r(U, (^) ,'jJ (U,^)) and ^

^

y:

form a sheaf on X.

Proof. The proof is routine. The first sheaf axiom is a consequence of Lemma 5.7.1.

The second sheaf axiom is a consequence of the definitions. D

Remark 1. The significance of the failure of the second sheaf axiom for basic open sets is that we cannot easily describe global sections T(X,^X),

X = Spec(A,^), or more generally

r(D(a),^x),

D(a) C Spec(A,'p)

a basic open set. Lemma 5.7.1 implies that

116

(AS(-a) ^ s ( a ) ) "* (r(D(a) ,0£ $ (D(a) ,^))

is

injective and that However, F(D(a),^Y ) A

'p g, . = A g , . n p ( D ( a ) , ^ ) may be bigger than

is the contracted order.

A c r .. b ^aj

In any case, one shows in

the usual way that there is a natural sheaf isomorphism

Remark 2.

It is perhaps good that

r"(X,/?Y)

is larger than

A

The natural map

(A,'£) -*- (Ac,,>. ,'.pcr,O oil) o^lj

In such case, T ( X , ^ )

Ac ^

is not always injective in (POR) .

may contain information about

(A,'p)

lost in the

localization In the category (PORCK), the second sheaf axiom for basic open sets, Lemma 5.7.2, does allow computation of the global sections of the structure sheaf.

We have

Proposition 5.8.4.

Let

(A/p) e (PORCK),

sections of the structure sheaf

X = Spec(A,'p).

€L over the basic open set

Then the D(a) C x

A

is the ring

F(D(a),^ x) = (A g . •x/^Sr ^ '

is the ring

T ( X , ^ ) = (Ag ( 1 ) ^ s ( 1 ) ) •

In

Particular»

tne

global sections D

The Zariski topology on the prime convex ideal spectrum of a partially ordered ring has the following property not shared by unordered rings. A finite union of basic open sets is still a basic open set. The proof is easy since D(f) = D(f 2 ) and D(IIf?).

U D(f?) = D(E f?), which is symmetrical with

H D(f?) =

If, further, our ring is Noetherian (more generally, if the radical

convex ideals satisfy the ascending chain condition),then every Zariski open set is a basic open set.

117

VI • Polynomials

6.1.

Polynomials as Functions Let

over

A

A.

be a ring, A | X ... X ]

Let

A ^

f e A [ X 1 ... X n ] If

(A/p)

A X , from for all

X

x e X,

1

If

to

(A, ]))

denote affine n-space over

defines a function

f:

A ^

A.

-» A

n-variables

Each polynomial by evaluation.

is a partially ordered ring, then so is the ring of functions A, for any set

X.

f (x) e '$. Note

belongs to

(PORCK)

injective?

Namely, define

A C AX

or

A very natural question is: -* A A

the polynomial ring in

'£x c A X

by

(the constants) and

(PORNN), then so does

f e$

^x

x

if

extends '£.

X

(A , £ x ) .

when is the ring homomorphism

A[X.^ ... X n ]

The following at least gives an easy sufficient condition.

Proposition 6.1.1.

If the ring

A

is

z (n) A f X ^ . .X^]-• A

Z-torsion free, then

is injective.

Proof.

We first establish the result for

a Q + a x X + • • • + adX

is a polynomial such that

in particular, f(0) = a

=0.

Next, let

equations in the coefficients

a.,...,a.

2 ad = 0

118

o

+---+da, = d

Suppose

f(k) = 0

k = l,2,...,d.

a. + a_ 1 2

da.. + d a 1 2

n = 1.

f(X) =

for all We obtain

k € Z C A. d

Since

2 d\ x, • • • x, \ x X 2

det

Xx

2

2

••• Xx d 2

x) n

, xx \ n

xx

2

n

n (x - x ) , l

(iii) (iv) (v)

2

{b|0£b £a 2

{b|f(b ) _ 0.

If

The following is proved

in general just as it is for real numbers.

Lemma 7.2.1(a). ordered field.

If

f (T) = a Q T n + a^ 1 1 " 1 + • • • + a R G E[T],

Let x,

b E E,

E

a totally

|x| £ |b|, then

|f(x)| < 2 |a. Mbl""1 . i=0 (b) If a € E

is a root of

f, then n

|a| ).

•

is positive, the polynomial

and positive if

T > 1 + p.

T -p

is certainly negative

From this fact, 7.1.2 and 7.2.3 we

obtain the "if" direction in the following characterization of real closed fields.

Proposition 7.2.4.

A totally ordered field

and only if for every polynomial

f(T) E R[T]

f(a) < 0 < f(b), there is a root

c E R

132

of

R

and f

is real closed if a,b G R

between

a

with and

b.

Proof:

The algebraic closure of a real closed

R

is

R[/T].

We

thus can factor

where the

r. E R

are the real roots and the

pairs of non-real roots. b

It is then clear that if

must lie on opposite sides of a real root

Remark: that if

a. _+ $. *^T

r.

are the conjugate

f(a) < 0 < f(b), a

of odd multiplicity

and

m..

•

The existence of this simple factorization also easily implies

f(T) G R[T]

f(T) = gj(T) + g*(T)

is nowhere negative on for suitable

g][ ,g 2

R,

R

real closed, then

e R[T].

Next we consider the geometric behavior of a polynomial function near a point

x = a, over any totally ordered field

f(x)-f(a) =

E.

f(x)

We can write

.,(a) (x-a) 1 .

I

Using the estimates of Lemma 7.2.1, one can verify the following.

Proposition 7.2.5.

Let

f(T) G E[T],

E

m ^ 1

be smallest such that

f ^ C a ) ^ 0,

a £ E.

in

such that:

E

(i) If

m

even, f

(a) > 0, then

(a - £, a + e ) , decreasing on (ii) If

m

odd, f

(a - e, a + e ) , negative on Moreover, on these positive when

f

a, the derivative

We obtain statements for the two cases when f(T)

by

-f(T).

f

(a,a+e).

is increasing on

and positive on

is increasing, negative when

e > 0

is non-negative on

and increasing on

f(x) - f(a)

(a-e,a)

e-intervals around

Then there is

f(x) - f(a)

(a-e,a)

^(a) > 0, then

totally ordered, and let

ff

(a,a+e). of

f

is

is decreasing.

fW(a) < 0

O

by replacing

We draw the four possibilities for local behavior below.

Combining this local behavior of polynomials with the characterization 7.2.4 of real closed fields, we can deduce certain global results about the behavior of polynomials over real closed fields.

133

i,

f ( m ) (a) > 0

m

fW(a) > 0

odd,

f• > 0

f' > 0 /

x= a

< 0

f1 > 0,

m

even,

vf' < 0

f ( m ) (a) < 0

m

odd,

f^(a)

< 0

Figure 7.2.6

Proposition 7.2.7. a < b

in

If

is real closed, f(T) G R [ T ] ,

R

R, then the derivative

More generally, for any

of

f

has a root

f(T) G R[T], there exists

*ir^

Proof:

f'

c,

c G (a,b)

f(a) = f(b) = 0, a < c < b. with

- f(b) -f(a)

The second statement (Mean Value Theorem) follows from the first

(Rolle's Theorem) in the usual way.

To prove Rolle's theorem, one first

assumes

a

f(T)

has no roots between

root closer to

a

if necessary.

and

b, by replacing

By 7.2.3, f

b

by another

does not change sign on

(a,b).

Then a short case-by-case consideration of the local possibilities for near

a

f'(a+e)

and and

b, drawn in Fig. 7.2.6, implies that for suitable f' (b - e)

Corollary 7.2.8.

134

have opposite sign.

Let

R

f

e > 0,

Now 7.2.4 completes the proof. •

be real closed, f(T) e R[T].

Then

f

is

monotonic on any interval [a,b],

closed interval the endpoints

a,b

(a,b) f

not containing roots of

f'.

On any

assumes maximum and minimum values, either at

or at roots of

f'

(a,b).

in

•

7.3. Sturm's Theorem Let

E

be a totally ordered field, f(T) G E[T]

polynomial.

A priori, the number, or even the existence, of roots of

in a real closure of

E

E

any real closed field extending

Proposition 7.3.1. E.

Let

for counting the real roots of

E.

uniqueness of the real closure of

real closures of

f

might depend on the choice of real closure.

theorem provides an algorithm in

over

an irreducible

E

Sturm's f(T)

in

We first show how this implies the

E.

be a totally ordered field,

R,,R~

Then there exists a unique homomorphism

two

R.^ -> R 2

E.

Proof: gives

Uniqueness follows from existence, since, first, symmetry

R 2 •> R,

over

over

E, since any

E.

But

R, •> R,

R,

admits no non-trivial endomorphisms

maps squares to squares, hence preserves order,

but also induces a permutation of the roots in

R,

of any polynomial over

E, which are, of course, finite in number. Sturm's theorem implies that if f(T) G E[T], then E(a) -> R 2 a

o

< a

l



D

+,

f((a,b)) > 0} .

is transcendental

'-P

, '^D

are, in

fact, total orders on R(t). Secondly, the case with

D

infinite is dealt with by replacing

t

by

1/t,

D(l/t) » 0, hence rational. Thirdly, suppose

D » a, as in case (ii), with

which we may assume monic, factor

f(t)

where

a. ± /-T 3-

(Recall

r.

are the real roots and R(/TF)

if and only if

over

is the algebraic closure of r. < a 1 —

in

R, if and only if

R.

a < t.

Given

f(t) G R[tJ,

Thus

are the non-real roots. R.)

Obviously, (t-r.) G £

(t - r.) G '$ 1

.

Thus

'p = '£

3-,+

3.,+

141

as asserted.

If

if and only if

t < a

in

'£, then

(t-r.) G'.]}

a,—

We sketch a second proof that

'$ = '$

if

+

a < t,

D = D R (t) » a.

f(t) = f(a) + f ^ (a)(t-a) + ••• + (f^(a)/n!)(t-a) n =

Begin by writing

(f^(a)/m!)(t-a) m (l + terms in (t-a)), for some In the abstract order

'^ on

small relative to

Thus the 'p-sign of

f^

R.

R[t], t - a

m > 0, with

f^

is positive, but infinitesimally f(t)

'£ = '$

, as asserted.

The case

is simply the sign in

t < a

We point out that the replacement of

R

t

is analogous.

by

1/t, converting behavior

of functions "at +_ «>" into behavior "to the right or left of formulated in terms of coefficients. n

a Q t + • • • + a n , then then

f 6'|5 T

if

n

f G ^

is even, a

> 0

odd,

D = D R (t)

and

closer to

D

f

R(t), then the polynomial some real field containing between

a

f

(a,b)

D(t) = - oo ,

One simply

R. and

By Sturm's algorithm, f

(t-ri) G $

r i G D, if and only if

A R C R(t)

in the case

R.

so we may assume

t G AR.

f.)

If

D

f(t)

QR C AR

is a non-trivial, convex prime ideal of D = D D (t) » a.

had

'£

on

and

b

Thus

in

'p = '$~.

transcendental can be

(t-r^ G $

Obviously,

D

-

be the valuation ring associated to

Conversely, assume If

a

(Simply

would therefore have

II (t-r.)((t-a.)2+ 3?). x i,j 3 ^

is infinite or rational, then

Archimedean over

(a,b).

b, contradicting our choices.

f(t) =

Finally, let

f G R[t], choose

would have a real root between

based on the factorization

if and only if

For each

in the abstract order

'p = '$

142

If

r

has constant sign on

A simpler proof that

a G R, hence

< 0.

than all the roots of

the opposite sign as its values on

D = D R (t)

R.

O

is transcendental.

choose

b

a

in

t .

such that the function

R

n

aQ > 0

f(t) =

n

a < D < b

a root in

or

0" is easily

D(t) = + oo , and

O

Next, suppose

a

Namely, if

if and only if

-oo

divides through by

some

(a) ^ 0.

(a). By the discussion of local behavior of functions of 7.2, we

deduce

If

r. < a,

(t-r.) G'p 1

of

if and only if

A R ± R(t)

and

A R f R(t). Either

R(t) t

or

is the maximal ideal, then

R C R(t). is non1/t G A R , Q R fl R[t]

R[t]. Necessarily, then, Q R n R[t] = ((t-a]), D

Corollary 7.5.2. Suppose

§!>•••»§!
0, 1 £ i £ k.

Extend the partial order to a total order and consider the

classification of total orders directly.

Corollary 7.5.3. Suppose on

E,

E

E

is order dense in

f(t) G E[t].

Proof:

Then

be totally ordered with real closure R

f(t)

and

f(t) < 0.

f(t)

f

is non-negative as a function

2 2 h f = Zg.

By 7.5.2, f(t)

(a,b) C R, hence

R.

is a sum of squares in E(t).

If there is no equation

be ordered with interval

Let

•

in

Eft], then

Eft]

can

is strictly negative on some

is negative at some point of

E, which

contradicts our assumption.

Remark. f(t)

•

It is known that if

f(t)

is actually a sum of squares in

is a sum of squares in

E(t), then

Eft]. See [ 4 ] , [ 5 ] , [ 6 ] for

detailed discussions of this point.

Corollary 7.5.4. (a) Let

E

be totally ordered with real closure

is ordered, non-Archimedean over residue field

A = A£/QE

R, and suppose

E, with valuation ring

A p C E(t).

is then a simple algebraic extension of

E(t) The

E,

A = E(a). (b) If in

t 6 A E , let

E(t). Then

Eft]

HEft],

g(t)€E[t]

with degree

g i (t) < degree

and only if

g(a) > 0

Proof:

f(t)

A = E[t]/(f(t)), and the ordering on

corresponds to a choice of a root of (c) If

Eft]

(f(t)) = Q

in

and

irreducible, A

f(t) > 0

induced by that of

f(t), a £ R.

g(t) = f(t)m (g Q(t) +g x (t)f(t) + ••• + g k (t)f k (t)),

f(t), g Q (t) i 0, then

g(t) > 0

in

E(t)

if

R.

Assertion (b) implies (a) since either

is a principal ideal domain, the localized ring

t

or

1/t £ A £ .

Eft].^^..

Since

is the

143

valuation ring

Ag, and

Q E = (f (t))E[t] , f , _ . .

This proves (b). Finally,

(c) is more or less obvious.

•

7.6. Total Orders and Signed Places Just as valuation rings in fields correspond to places, we will show that totally ordered valuation rings with convex maximal ideal correspond to a special type of place, in which we distinguish place on a field

F

+ °° and

immediately yields a total order of

totally ordered valuation ring in

- °° .

A signed

F, in fact, a

F, with convex maximal ideal, and con-

versely, such a valuation ring yields an equivalence class of signed places. Let to

A

A

be a totally ordered field.

We adjoin symbols

- ©o

and extend the operations of addition, subtraction, multiplication,

and division between

°°, - 0

I - 00 ifi

a < 0

r

a • (-00)

144

oo and

oo 00

r-00 = j u 00

if

a > 0

if

a < 0

as follows.

(These rules

Division

a/°o = a/-00= 0

o°/a = (l/a)« 00

0* ( ± ° ° ) , ± °°/0,

the four possibilities

a ^ 0

if

a / 0

(l/a)-(- 00)

- 00/a =

The symbols

if

00 + (_ 00),

00 - 00 ,

± 00/+ 00 are undefined.

(- 00) - (- 00)

and all of

Of course, we want the usual

commutative and associative rules to hold, in all defined expressions. that we do not distinguish the signs of both

0. If

a/- °°, a ^ 0, but call

These elements can be distinguished by "inverting". K

function and

a/°° and

is a field by a signed place, with the values in A, we mean a

p:

K->A, ± °°, such that

p(x+y) = p(x)+p(y),

p(l) = 1, whenever the terms are all defined.

guarantees

p ^ 0

and

the ring

A

Let

p:

= {x G K|p(x) G A}

The last condition simply

K -• A, ± 00 be a signed place on is a valuation ring.

Moreover, K

ordered if the strictly positive elements are defined to be p(l/x) = °° , or

p(x) > 0

the maximal ideal

Proof: p:

Q

in

C A

A}.

If the order on

products

K

x, -x G p . xy

and

becomes a place on

K.

It remains to show that

x + y, for

The possibilities for

Then

is totally

'£ = {xEK|p(x)= °°,

is restricted to

First, if we simply suppress the distinction between K -* A, 00

K.

A ,

is convex.

Thus

Secondly it is clear that for all non-zero one of

p(xy) = p(x)p(y),

p ^ °° .

Proposition 7.6.1.

- °° ,

Note

A

is a valuation ring.

x G K,

'£

+ °° and

x

G '.p+, and exactly

is closed under sums and

x,y G ']> .

p(x), p(y)

are covered by (i) p(x),p(y) > 0,

(ii) p(x) = 00 , p (y) > 0, (iii) p(i) = °° , p(y) > 0, (iv) p(x) = °o , p(y) = 00 , (v) p(-) = 00 , p(i) = 00 , (vi) p(x) = 00 , p(I) =00. it is trivial to check directly that evaluate either p(xy) ^ 0.

If

p(xy)

or

p(xy) < 0

This contradiction shows

p(l/xy) or

xy G'|) . directly.

In case (vi), one cannot By symmetry, we may assume

p(xy) = - «> , then

p(xy) > 0

or

i n all cases except (vi)

p(x) = p(xy y) = p(xy)p(-) =

p(xy) = °° .

The six cases for sums

145

are all trivial except (v). But

x + y = xy(— + — ) , x y by product cases, and case (iv) for sums. Finally, Q homomorphism

C A

A

is convex as the kernel of an order preserving

-*• A.

A signed place

•

p:

K -• A, ± °° determines a total order on

given a valuation ring ideal

Specifically, if

in a totally ordered field

1/x £ Q, then we set

and

p(x) = - °° if

and

p(xy) = p(x)p(y)

with in

A

Q C A, one recovers a signed place

that since

A

x

is negative.

This shows that if

Verification of

ap^

Here

valuation ring of

and

p^:

p i (K)

p..

valuation rings in

K -• A^, ± °° ,

Remark 7.6.2.

g:

K

P-^A

A

is all of

K.

maximal ideal, one sets becomes the "addition" in

of

K,

K

G Q.

Thus

) C A ^ where A

C K is the

and totally ordered A signed place

p

of

K

K.

correspond to a third concept, that of

A valuation is a function

If

A C K

T = K*/A*, T

and

A - (0) = {x G K|v(x) :> 0}

A* = A - Q. 1 E K*

and

v:

K* -* V, where

K* = K - (0), with

is a valuation ring with

The natural projection

valuation, v(x) = v(-x).

146

0 _ min(v(x) ,v(y)).

if

1

p-^ (K), ± ° ° ^ p 2 ( K ) , ± °° such

a totally ordered (additive) abelian group and

x A , °°+ A, °° corresponding to the pair of valuation rings

A Ck,^

149

in

K.

But see the Remark following the proof of the next proposition.)

Proposition 7.7.2. (Krull) with valuation ring

(A,Q).

p H A - convex ideal, and

Let

Let

']) C K

$" = 'p/Q

be any total order refinement of p! 3 $

refinements

on

K

Moreover, all prime ideals

Proof:

Let

Q~

pf:

and by

A = A/Q

such that P CA

a:

Let

K,

Q

a

p"' D f

Then there exist total order 'p1 n A - convex

is

p'/Q=Tf-

and

p' n A - convex.

1/x G K

K* -> {± 1}, then refine

p:

with

x G Q.

K -> A, °° to a

'JT1 C A, by p 1 (x) = p(x), x E A

K •> A, ± °° , relative to The order

f

CK

will be the order induced

p'. A*

denote

x = yz, with

product

A - Q, the units in

z E A*.

x-j^-.-'x, ,

A subset

x^ G s,

possibly

a:

then there is a relation xi

define

occur.)

Define

a(y) G {± i}

are fovoed on us if orders on

p

x,y G K*

are associates

'p -independent if no finite

'p - independent subsets

SCR*

although

S H A* = 0.

S •> {± 1}

a(z) £ {± 1}

be an arbitrary function. x. G S , p £ j ) ,

to be the sign of

z

If

y G K* - S,

z G A*.

(Possibly

(A, p"1)

in

and

cr(y)a(x1) • •-a(xk) = a(z). The point is, these choices is to refine

p

and if we are to have

p'/Q = 'p!

as

A.

We first check that yx| .. • x^ = p'z an associate of choice of

S.

1

with

a:

Thus

K* •* {± 1}

x! G S,

pp'/y

and it follows that

p. !

verify that

p'

Verification of

Gp,

is well defined. z

1

G A * , then

This can occur only if f

pz = p z , cr(y)

p

1

f

z = (p'/p)z ,

If also x-^ • -\*[- • - ^

{x^} = {x!}

(p'/p) G A * , hence

pf:

a(z)=a(z'),

K •> A, ± °° and we only need

is a signed place, relative to the total order f

p ' O y ) = p (x)p'(y)

is

by our

is independent of the choices made.

At this point we have a function

150

is

v x ^ — x ^ = pz, with

by 1

We say

x^ ^ x., is an associate of an element of 'p.

S = 0, and certainly

In any event, let

A.

S C K*

By Zorn's lemma, there exist maximal

no

K, with

the induced order.

A. Q

are

be a place of a field

be a partial order on

p" on

p 1 (y) = a(y) • °°, y E Q" 1 .

Let if

c

K -> A, «

denote the subset of elements

We will define a function signed place

p:

•

'p' on

is quite simple and verification of

A.

p'(x+y) = p 1 (x) +p f (y) Finally, let

is also easy if use is made of the identity

P CA

signed place

p : K -> A, ± °° to

The positive elements in in

A. p . C K.

A p = A, p ./P.

Restriction of the

induces a signed place

A p -* A, ± °° .

are clearly the images of the positive elements P c A, p >.

|5f n A. p . -convex.

is

•

The last paragraph in the proof can be established more directly.

In fact, if

(A,Q)

f = '£d

Q

and

is a valuation ring, '£ C A

-^-convex, then any ideal

Namely, suppose is

Ap

A, p . C K

But this just says

Remark:

Q

be a prime ideal,

1

x+y = x ( l + — ) .

0 A, °° , where

E, we obtain orderings of

A.

-p-independent set will consist

Thus there are exactly two orderings on

A, determined by whether

an

f(x) £ K

K

over

is made positive or negative.

We turn now to our second main existence theorem, the construction of signed places on a field

Lemma 7.7.3. P C A

a prime

'£' D'$

on

K

Proof:

Let

K

with preassigned center on a subring

(K, £)

be a partially ordered field, A C K

'£ n A - convex ideal. such that

P

A C K.

is

a subring

Then there exist total order refinements

tp1 H A - c o n v e x .

We use the basic result of

A.

Klapper, Proposition 2.7.1, that

if a prime ideal is convex for an intersection of two orders, it is convex for one of them separately. p[x]

and

since if y £•£.

'p[-x]

x £ K,

are (derived) orders on !

!

y = p + qx = p -q x, Thus our prime ideal

^P[-x] n A-convex.

If

x, -x £ p , then both simple refinements K.

Moreover,

P>p'>q,q' ^'P> then P CA

is either

p[x] H^[-x] = '$

(q+q')y = pq' + p f q, hence

'.p[x] O A - convex or

Zorn's lemma now implies 7.7.3.

•

151

Proposition 7.7.4. subring, order

PCA

a

Let

(K,'£)

be a totally ordered field,

'$ n A - convex prime ideal

1

Q , such that

Af = A'/Q',

We may as well assume

by passing to /HPA[x]

or

assuming

1

A C A ,

(A,p. , PA, .) /HPA[l/x]

x > 0.

m

A

a^jb.

e

111 1

P,

m )

convex:

P./P. CA/P. ji

is

i

Proposition 7.7.9.

( • • • ( f f / P J . / P , ) -1• - / P .1) °

d

d

d

- convex.

In the notation above, the conditions (*) are

necessary and sufficient for the existence of a total order refinement $' D'P

on

K

Proof: orders on

such that all

Let K.

P.CA

x E K, with

x, - x £ ' p .

As in the proof of 7.7.3,

generally that if

if

'£, '£', tp

that conditions (*) hold for

p ! HA-convex.

are

Let

'P'=1>[x],

']>' 0 $ " = f.

are orders on

A

with

'£ M =1>[-x]

as

Now we will prove f

'P n •£" = $

'$, then conditions (*) hold for

and such

'£' or for '£".

The proposition then follows from Zorn's lemma. We distinguish two cases, case (I) if case (II) if

P

is both

conditions (*) hold for with prime ideals

'£' and

P

'^"-convex.

CA

'£'. In case (II), we will pass to the ring

P i /P Q = P±, and orders

to the chain of prime ideals $'

or

'£" on

number of prime ideals on

A = A/P Q .

"^"-convex, and

In case (I) we will prove

f' = C P 7 P O ) , , and

Then we will establish conditions (*) for the order

for either

is not

P^

A

P.,

1 £ i 0} Z(g) = {x G R n|g(x) = 0} .

If

g. G R[X •••X ]

is a finite collection of polynomials, define

u{ gi } = nu(g.) w{ gi} = n w( gi ) z{g.} = nz(g.) . Any semi-algebraic set sections of

(*)

Zf s

E C R

and

can be expressed as a finite union of inter-

U's

E = u (Z{fik> n u { g j k } ) ,

for finite collections of polynomials is of course highly non-unique. properties of the sets tations (*) .

162

E

f-v> S-v*

Such a representation of

We are primarily interested in "geometric"

themselves, not in the particular represen-

E

Finite unions of the Their complements in

R

n

U{g^}

will be called open semi-algebraic sets.

will be called closed semi-algebraic sets.

typical closed semi-algebraic set

S = U si ,

S

A

can be represented

where

(**)

f..} n w { g . k } .

Of course, equalities and

f = 0

could be avoided altogether by writing

f > 0

f £ 0, but this is psychologically less natural.

Lemma 8.1.1.

If x G U{g i >, then there is

g G RfX^.X^

such that

x 6u(g) Cu{g.}.

Proof:

The estimates of 7.2 can be used to find a ball around the

point

x = (x.....x ) G R n , contained in

U{g.}.

where

B(x,e) = {y| lly-xll < e} ,

= £(y.-x.) .

given by

2

Hy-xll

That is, B(x,e) C u { g . } , A suitable

Rn.

topology on

•

U(g)

form a base for a

But we now emphasize that semi-algebraic geometry is

precisely not concerned with this strong topology. ring of semi-algebraic subsets of

R

Instead, only the Boolean

is relevant.

If

E CR

is semi-

algebraic, then we also get a natural Boolean ring of subsets of by intersecting

closure

is then

g x > £ (y) = e - E ( 7 j - X j ) .

Of course, Lemma 8.1.1 states that the sets

If

g

2

E, F ¥~

of

E

with other semi-algebraic sets.

are semi-algebraic and F

in

E

F° = {x e E | exists

F"E = {x € E | for all

Of course, F°

E, simply

and

F

in the topology on

E

F C E, define the interior

F°

and

as follows.

e > 0

such that

y G E, lly-xll < e

e > 0, there exists

yEF

implies

such that

yEF}

lly-xll < e}.

are the ordinary toplogical interior and closure of with base the sets

U(g) H E .

F

But, for our purposes,

the point is that it follows easily from the Tarski-Seidenberg theorem, to be

163

discussed below, that we will simply write

F° F°

and and

F

are also semi-algebraic.

(If

E = Rn,

F.)

We now make the following confusing (but crucial for understanding the differences between algebra and topology) definition. F C E F

A semi-algebraic subset

will be called an open, semi-algebraic subset of

will be called a closed, semi-algebraic subset of

commas should not be ignored, at least for now.

E

of

U{g.} n E

E.

if

if

F = Fp

F = F .

and The

It is easy to check that

the complements of the open, semi-algebraic subsets of semi-algebraic subsets of

E

E

are the closed,

It is also easy to check that finite unions

are open, semi-algebraic and (hence) finite unions of

Z{f.} 0, where the

P.

are polynomials over a real

closed field, using basic logical connectives "and", "or", "not", and quantifiers "exists

164

x.", "for all

x.".

The decision procedure amounts to checking

whether or not certain polynomial inequalities involving the coefficients of the

Pi

hold.

For example, the sentence "there exists

2

x G R

such that

2

ax + bx + c = 0"

is equivalent to

"b - 4ac >_ 0".

Also, Sturm's theorem in 7.3

is a special case of such a decision procedure. The proof of the Tarski-Seidenberg theorem is not difficult. Cohen's proof [62] in an appendix.

We will give

From the algebraists point of view, what

is involved is just an argument making use of (1) induction on degrees of polynomials and number of variables and (2) explicit calculations in polynomial rings involving partial derivatives and division algorithms.

In other words,

elimination theory. The applications of the theorem are rather striking. sharp distinction between two types of application.

We will make a

The first type, which

is almost a reformulation of the theorem itself, says that any set defined in terms of semi-algebraic sets by an elementary sentence is still a semialgebraic set.

For example, this includes closures and interiors mentioned

above, images of semi-algebraic sets under polynomial mappings and other frequently used constructions.

R n -> H m ,

(However, the theorem gives little

insight into the question of whether a set is, say, open.)

It is amusing that

even the simplest special cases of this type (say the projection to zeros in

R

R

of the

of a single polynomial, or the closure of the set where a

single polynomial is strictly positive) are no easier to analyze than the whole theorem.

Thus the Tarski-Seidenberg theorem is a very efficient tool.

We emphasize that this type of application actually provides a proof that the asserted set is semi-algebraic, simultaneously for all real closed fields, in fact, an elementary proof.

The reason is, any single elementary sentence

is just a special case of the theorem.

The more subtle application is this.

Given an elementary sentence, suppose it can be checked in one real closed field where it makes sense and

is true.

For example, it might be checked for

the classical real numbers by transcendental methods (use of completeness, possibility of integration, etc.)

Then the sentence must be true for any

real closed field to which it applies, because there exists a decision procedure which is independent of the field in which it is carried out. Thus, one might say that this method of application amounts to proving by non-

165

elementary methods in one special case that an elementary proof of the statement in general does exist.

It certainly might be very tedious, if not

physically impossible, to actually work out this elementary proof. In this book we absolutely and unequivocally refuse to give proofs of this second type. fields.

Every result is proved uniformly for all real closed ground

Our philosophical objection to transcendental proofs is that they may

logically prove a result but they do not explain it, except for the special case of real numbers.

Also, one of our central themes is that the real numbers are

totally irrelevant in algebraic topology, so it would not do to rely on them at some point in our chain of reasoning.

Finally, there is already a respect-

able tradition in this century of finding non-transcendental proofs of purely algebraic results concerning algebraically closed fields.

We think real closed

fields deserve (at least) equal time and effort. We do not at all mean that only elementary proofs are acceptable. we use Dedekind cuts, total orders, and signed places repeatedly.

In fact,

The point is,

in the form we use these concepts they apply uniformly to all real closed fields. One advantage to developing such techniques is precisely that one is not tied down to "elementary sentences".

There is rich non-elementary theory to be

studied for arbitrary real closed fields, and even if a statement turns out to be equivalent

to

an elementary statement, it may be unnatural to dwell

on this fact, and even worse to be forced to depend upon it. There are obviously aesthetic questions involved in this discussion.

We

admit that many of our proofs are long and could be replaced.by the single phrase "Tarski-Seidenberg and true for real numbers". effort is worthwhile.

However, we feel the

In fact, we do not use the Tarski-Seidenberg theorem at

all, until 8.7. We now change tacks somewhat and indicate a more invariant approach to semi-algebraic geometry. R-algebra of finite type.

Suppose

(A,p) £ (PORNN), and assume

Define a set

X = X(A,'£)

A

is an

by

X(A/p) = H6m ( p 0 R ) ((A,tp),(R^ w )).

(Of course, 'C

166

is the only order on

R.)

We have the canonical adjoint

homomorphism from the elements

A

to the ring

R

of

R-valued functions on

f £ 'p define functions nowhere negative on U{g i >, W{g.}, Z{g.} C X

the subsets

X.

X, and If

g. E A,

are defined in the obvious way. Interiors

and closures of subsets of X are also defined just as above, essentially using the topology X with base the collection of sets U(g), g^A. (The U(g), g e A, are invariants of (A,'£) and replace the e-balls in the earlier definition.) If we choose a specific presentation I CRpC^.-.X ]

A = R[X,...X ]/I, where

is some ideal (necessarily radical and

n

gets identified with a subset of the zeros of

I, Z(I) C R .

X = X(A/p) = {x G Z(I) |g(x) ^ 0 all

Obviously, X

is a closed subset of

Rn

$ -convex), then

X

Specifically,

g € tp}.

in the topological sense. y

Now, the problem is, the homomorphism In fact, X

could be empty.

Definition.

RHJ-algebra (real, Hilbert-Jacobson) vif

is an R-algebra of finite type P C A , and

f £P

and

need not be injective.

Thus, we make the following definition.

(A,'£) is an

ideal

A -• R

A

(A,'£) £ (PORNN), and for each prime 'p-convex

g £ P, there exists

x E X

with

f(x) = 0

for all

g(x) t 0.

Corollary 8.1.3.

If

(a) the homomorphism (b) the set

X

(A,'£) is an A -* R

RHJ-algebra, then

is injective

is identified with the maximal convex ideal spectrum of

(A,?)

(c) for any subset

J C A, we have

vtt(J/p) = {£ e A|f(x) = 0 all

where

Z(J) C X

is the set of zeros of

xC-Z(J)},

J.

Proof: (a) The intersection of all '^-convex prime ideals of (A,'p) G (PORNN).

167

A

is

(0), since

(b) Each all

e

f '£.

x £ X

corresponds to a surjection

Thus kernel (p) C A

the definition of

contain

/H(J,$) J.

f €= A

is already maximal, then

"jj-convex maximal ideals containing

which vanish on all zeros of

J

in

P.

Thus

P

is /H(J,$)

X.

D

RHJ-algebra is attractive, there is still a

Without some control on the order

p C A, the resulting subsets

of affine space (relative to a presentation

can be rather chaotic. an

P

P

x £ X.

Although the notion of

X = X(A,p)

Conversely,

Again, the definition and (b) says each p-convex prime

is the set of

p(£) >^ 0,

is the intersection of the prime '^-convex ideals which

the intersection of all

problem.

with

RHJ-algebra states that every '^-convex prime ideal

corresponds to some (c)

A -*• R

is a maximal, '^-convex ideal.

is contained in such a maximal ideal. Thus if P

p:

A = R[XX ...X n ]/I)

The control needed in order to guarantee that such

X(A,'£) is semi-algebraic (necessarily closed, semi-algebraic) roughly

amounts to finiteness conditions on orders naturally associated to •£. Our study of semi-algebraic geometry will essentially amount, then, to identifying a large class of

RHJ-algebras

(A/p) with

X(A,'p) semi-algebraic and,

conversely, given a closed, semi-algebraic set RHJ-algebras easy if

E

(A,'p) with

E = X(A,'£).

E C R n , construct natural

This last turns out to be fairly

is closed semi-algebraic, but not so easy if

E

is only closed,

semi-algebraic.

8.2. Some Properties of RHJ-Algebras In this section we will construct many

RHJ-algebras, assuming the

following basic real Nullstellensatz, which will be proved in 8.4.

Proposition 8.2.1. is an

The polynomial ring with the weak order

(R[X^.. .Xn] ,'p

RHJ-algebra.

Of course, X(R[X,...X ] ,'£ ) = R n . l n w

•

Once we have this one

RHJ-algebra,

there are natural constructions of others. We indicate several such constructions in the propositions below.

168

Proposition 8.2.2. 'P(X) = { f G A|f (x) :> 0

If

(A/-P)

all

is an

RHJ-algebra

x G X}, then for any order 1

$ O p ' C ^ ( X ) , we have that

(A,? )

In particular, the radical

is an

X = X(A/P), and f

CA

with

RHJ-algebra and

X(A/p») = X(A/P).

'p-convex ideals and the radical

'.p1-convex ideals

coincide.

Proof:

Any

'£'-convex ideal is also 'p-convex.

obviously define

'-P(X)-convex ideals, hence also

Since the points

x ۥ X

f

'p -convex ideals, the

proposition follows easily.

Corollary 8.2.3.

•

If

(A/p)

and

(A/PJ m are RHJ-algebras. X(A,!Pm), and $m

Proof: P

=

'-P(x)

is an RHJ-algebra, then

(A/p ) , (A,'P ) ,

We have X = X(A,tp) = X(A/PJ s = X(A,:pj p =

The orders

and

it

'£ , 'p , :p were defined in 3.12. r s rp rm eas Y t 0 check that for an RHJ-algebra

is

By definition, J ' (A/p), we have

• Proposition 8.2.4.

If

(A,'p.)

are

RHJ-algebras, 1 £ i £ k, then

(A, H'p.) is an RHJ-algebra and X(A, rvp.) =iUX(A,!p.). l I l Proof:

The results of 2.1 show that any

'.p.-convex for some

j.

The result follows easily.

Proposition 8.2.5.

Proof: P C A

Any prime

is a prime

(A,'p )

Proof:

is an

Write

(A,'p)

is an

RHJ-algebra and

'^/I-convex ideal of

If

A

•

RHJ-algebra and

'.p-convex ideal containing

Corollary 8.2.6. then

If

(A/I, (]5/I) is an

'^-convex ideal, then

H *p. -convex prime ideal is

A/I

I CA

is a radical,

X(A/I, -p/I) =

is of the form

P/I, where

I.

D

is a reduced, real R-algebra of finite type,

RHJ-algebra.

A = R f X ^ . .X n ]/I, where

I CRj^.-.X^

is a radical,

169

'P -convex ideal.

(This is exactly what the hypotheses on

A

mean.)

Now

apply 8.2.1 and 8.2.5.

•

Proposition 8.2.7. such that

A

is an

is finitely generated over n

(A/£ A ), where

over

(B,£B)

If

£ A = A '-Pg, then

X(A,'£A) = image (X(B,$B))

R

and

(A,'£A)

A C B

(B,'£ )

is an

under the natural map

induced by the inclusion

Proof:

RHJ-algebra,

a subring

is semi-integral

RHJ-algebra and

Spec(B,'^B) -• Spec(A/pA)

A C B.

This is immediate from the going up theorem for semi-integral

extensions proved in 6.4.

Proposition 8.2.8. let

f CA

Let

A

(0) = O p . ,

Then

Proof:

(For example, the

(A.,'p.)

Next, suppose g £ Q^.

P i C Q, and g(x) t 0. Thus

A

is an

(A,'£)

Then

g £ Q.

Then

(Ai,']5i)

Qi Find

A

contains some

is an

is an

x G X(A,'£) x

Proposition 8.2.9. Pi C A

given orders

Let

A

by 3.11.)

A -*• R P-.)

under the inclusion

(A^p^) = is an

RHJ-algebra

contains some

This implies

Q^ C A.

is a

^-convex prime

so that

x

P. .

X(A,'p) =

'.{K-convex

is a zero of

Q

and

g(x) t 0.

The converse is equally routine, using A

contains some

P^.

•

be a reduced R-algebra of finite type,

and define an .order

(0) = n p . , '|5 C A

by

Suppose

'|5 = A f l l f - ,

A -• n A. . Then X(A/p) = U X(A. ,'p.) and (A,tp) l l l i RHJ-algebra if and only if each (A^^K) is an RHJ-algebra.

170

(In

Q C A , with

Q i = Q / P ^ and

be a finite collection of primes with !p. C A . = A/P.

Let

Moreover, (A/P)

is a zero of

RHJ-algebra.

the fact that any prime ideal of

could be the minimal primes

RHJ-algebra and

corresponds to a

x e X(Ai>^i),

P.

'p-convex

RHJ-algebra.

The kernel of any homomorphism

fact, any prime ideal of

prime

be any finite collection of

X(A/P) = U X(A.;j5.).

if and only if each

an

P. C A

A, which are convex for any order on

(A/Pi, '-P/Pp.

let

be a reduced, real R-algebra of finite type,

be an order, and let

primes with of

•

is

Proof: for some

By Proposition 2.7.8, if

i,

P. C Q

and

Q/P.

is

Q C A

is any

$.-convex.

'^-convex prime, then

All parts of the proposition

above follow routinely from this fact.

Proposition 8.2.10. and

'£ = '£ [g.] w j

is an

A

is a reduced, real R-algebra of finite type

is a finite refinement of the weak order on

A, then

(A/p)

RHJ-algebra.

Proof: refinement

Let

P. C A

'p [g.] = 'p. I w j

Of course, some •}K.

If

•

Now, A^

g.

be the minimal primes

will also be an order on

may be

0

in

A.

B

.

is a domain, unless

g

1

§m+1

and go on to

steps, we have a domain

B

= A.,

where

*-Pw(B) C B

is the weak order.

Applying 8.2.7, (Ai,tpi)

(A,p)

RHJ-algebra.

Remark.

g

. = B [/g

. ] , m > 0. B .

The first

, £'J> [!•••••§ ] C A . .

instead.

B = lim B , integral over

RHJ-algebra. is an

B

g. ^ 0, consider the

is already a square in

Bm[>/gm+2]

R, and real, say be Proposition 6.2.1.

g. = g. (mod P . ) . j j i

and these can be omitted in studying

time this happens, an easy computation shows we can skip

The finite

A., where l

is an integral domain, and assuming all

finite sequence of integral extensions Each

A. = A/P..

Thus,

After finitely many A., of finite type over

Moreover, f. = 'p [g.] = A. n •£ (B), Applying 8.2.6, (B,$w(B)) is an RHJ-algebra.

is an

Finally, by 8.2.8, •

The proof of the basic Nullstellensatz 8.2.1, to be given in

8.4, will, in fact, yield directly that any order of type

finite real integral domain over R gives an

'£ [g-] w 3

RHJ-algebra.

be used instead of 8.2.7 in the proof of 8.2.10.

on a

This could then

On the other hand, both

proofs use the device of adjoining square roots. If

(A/P)

is an

X(A,'p) C X(A,? w ). Now, if

RHJ-algebra, then so is

In fact, X(A/£) = {x € X(A,? w ) |g(x) > 0, all

A = R[X1...X ]/J, where

^-convex ideal, then n

Z{f.} C R . 1

then

If

(A/p w ), by 8.2.6, and

X(A,'^w)

£ = ^ [g.] W

J = (f/) C R p ^ . J ]

g

is a radical,

is identified with the real algebraic set

is a finite refinement of the weak order on

J

X(A,'£) = Z{f.} Hw{g.}. 1 3

More generally, if

!p = H •£K

where each

171

A,

$

= 'P w [g- k ]

is a finite refinement of the weak order, 1 £ k £ m, then by

8.2.4, X(A,p) = U z{f i > n W{g. k >.

Thus orders of this type give rise to

closed semi-algebraic sets. We now want to prove that any closed semi-algebraic set

(**)

s = us.,

can be represented as A

a quotient of

RfX^.-X^. S.

Trivially, I(S) = n I(S i ), S.,

is an order on Let

P.

A ^ , since the

We now have

k(S/) = R[X X . . .X J/ICS^ g.v

g-k

A i a = R[Xr..Xn]/Pia.

I(S.)

Then

f±a

in

P.

A(S.)-

= » w [g i k l

P i a C P.0

implies

and thus we have an inclusion

P.

CP.»

A(S)

occur among the

hold, even

are the minimal primes of

RHJ-algebra and

Proof:

we have

I(S.), we know

If

$ = A(S) n

n

^

C A ( S ) , then

(A(S),^)

is

X(A(S),'^5) = S C R n .

X(A(S),tp) =

X(A(S),'^) C S.

I(S.)

is a zero of

as an

A. -ideal, since to ik ia

U

*(\a>Via)'

Conversely, since

P. , some

a.

g., (x) > 0. -

If

Since clearly

x

Of course, the order

is an

XCA^,^)

I(Si) = C\ P ^ , any zero

x G S., then

Thus

(A(S),'P)

is clearly

x

c S

'^ -convex

•

y

p C A(S)

constructed above depends on the

i>

of

S. C U X(A. ,'£. ) , and l a ia la

S C X(A(S)/P).

172

= P.^,

1

Propositions 8.2.9 and 8.2.10 guarantee that

RHJ-algebra, with

P.

i i j.

Proposition 8.2.11. an

is an order

RfX^^.. .X n]/I(Si ) = A(S^).

10L

that

IK

R|X . . .X ] , so that

are convex for any order on

It is possible that some relations i, the

$ T[g.,] W

may be zero in

Note that all the minimal primes of

but since for fixed

S., hence 1

Of course, some

P.a/I(S^)

S.

is a ring of functions

are non-negative on

I (S) = H I(S.) = flflP.

A(S) -> II A. . P. .

is a radical, f -convex ideal

denote the minimal primes over Let

denote the ideal of

IK

A(S.)«

I(S i ) = n PiQj. on

I(S)

(A,'p), with

is a ring of R-valued functions on

1

1J

RHJ-algebra

I(S) CR[X 1 ...X n ]

Then

f.. £ I(S.), and the

1

for a suitable

Let

A(S) = R[X,...X ]/I(S)

since

on

s i = z{f..} nw{g ik }

S = X(A,'.p)

functions which vanish on

S C R ,

representation (**)

of

{f GA(S)|f(x) > 0

all

RHJ-algebra. S C Rn.

S, and is a rather weak order. xGS}.

We refer to

'p(S) =

Then by 8.2.2, (A(S)/P(S))

(A(S) ,'£(S))

The fact that it is an

Let

is also an

as the affine coordinate ring of

RHJ-algebra is geometrically very satisfying.

However, a useful question is, how does one go about deciding algebraically, in terms of a representation (**) to

'P(S)?

f G'-P(X')

for some

n >_ 0,

this implies

f G A(S)

(A',p'), with

RHJ-algebras

if and only if there is an equation

p,q G ^ 1 .

belongs

X' = X(A',p'), (f 2 n + p)f = q,

(The "if" part is trivial.)

'£' = $(X f ) = '£' , where

defined in 3.12. f

S, whether a given

One answer is provided by a theorem of Stengle, which asserts

that for a certain class of we have

of

'£' and

In particular,

'£' are the operators on orders

In fact, it also shows that in this case the collection of

satisfying such a formula is closed under sums, thus simplifying the con-

struction of

p1

in these cases.

The most general class of orders satisfying

Stengle's theorem is obscure, but it includes at least all is a reduced R-algebra of finite type, and

f CA

(A/p), where

A

is a finite refinement of

the weak order. In particular, reconsider the closed semi-algebraic set by (**)

above.

R[X r ..X n ]/P i £ x ,

We constructed an inclusion P.a

prime.

S = X(A(S),A(S) H n 'Picp-

A(S) -• II k^a>

We had the orders

^

S

described

where

A^a =

= 1>w[gikl C A.^

and

As a consequence of Stengle's theorem, we can

now state

Corollary 8.2.12. $(S) = A(S) n nCP- ) , where .

ia

CPia) = {f e

AicJ(f2n+

p

p)f=q, some n > 0 , p , q 6 y .

D

This result characterizes algebraically the collection of polynomials f G R[X 1 ...X n ]

non-negative on the closed semi-algebraic set

S C Rn .

We

will prove Stengle's theorem in 8.5. Note that recovering

S

from the

(A. ,'p. )

an algebraic set from its irreducible components. setting, where orders

f^a

is analogous to recovering In this semi-algebraic

are carried along with the integral domains

173

A i a , or with the prime ideals

P.^ C R[X^...X ] , we can have the same

occurring several times with different orders

'£ [g-k]

corresponds to several patches on the variety

Z(P.^)

W

^g'k^

n

Z

( P icP J

P

ia5 P j0 J ° r

a11

contained in

e< uivalentl

l

belonging to

S.

Remark.

with

P a t c h e s on

The orders

$ [g.,] W

on

A.

IK

so we can find

V w [ g i k | g i k £ Pia] •

x G X(\a$ia)

hence the patches

^ia*

z p

( j^)

= R[X,...X ]/P.

XCc

Let

x

n

g =

also

g(x) ^ 0.

)

contain points

are strict, g i k (x) > 0, for all tJie c o n

d

ition

g^k >. °

is

g ik -

Then

g £

P^,

ik ^ P ia

with

W{g.^} H Z ( f

considered above

10c

II g

ik

of the form

nw( gl )

can be replaced by

e

This

We can even have proper inclusions

y» Z(?j^ $ Z ( P i c P '

— — — —

g i k (x) :> 0

A*/y"

P^a

S.

s=

g

on

It follows that x

where the inequalities

gik ^ P ^ .

superfluous on

z

g i k 0 0 > 0,

Of course, if

P

( £a)-

We have indicated above how certain operations on orders defining RHJ-algebras yield other

RHJ-algebras.

Certain other operators, which one

might hope do the same, turn out to be more subtle. are

RHJ-algebras

A

is

( /P[g])>

(A,'p)

such that some simple refinement of the order '^,

not an RHJ-algebra.

i: A -* A[t]

where

indeterminate over

We will show that there

Also, (A[t],iJ>)

is the inclusion of

A

need not be an RHJ-algebra,

in the polynomial ring in one

A.

The basic example, which is a good source of counterexamples, is (A,p) = (R[x,y], ? w [x] n ? w [ y ] ) . Suppose

h(x,y)

Thus, X(A,?) = {(x,y) G R 2 |x > 0

is nonnegative on

X(A,'^).

Since

h

or

y > 0}.

cannot change sign

across either the positive x-axis or the positive y-axis, it follows that

174

h(x,y)

is divisible by an even power of

for some

r < 0,

{(x,0)|x < r},

h

x

-x,-y

y.

It then follows that

is nonnegative on a neighborhood in

{(0,y)|y < r}

on the negative

A somewhat surprising consequence is that is an order.

and

This follows since for any

x-

and

R

of the two rays

y-axes, respectively.

$[-x,-y] = CPw[x] n •£ [y])[-x,-y] Si >•••>&]< *= P> the elements

are simultaneously positive on a non-empty open set in

sets in

R

R .

are Zariski-dense, relative to the ring of polynomials

g.,

(Open R[X,...X ] ,

because of the finite Taylor expansion of a polynomial about any point. holds

for any real closed

This

R, since the estimates of 7.2 show that the

formal algebraic partial derivatives of a polynomial agree with the "limit" definition of partial derivatives.) It is unclear whether or not

(A,'.]S')

2

X(A,'^') = {(x,y) G R |x < 0 < y}. be an

RHJ-algebra.

cannot be an

is an

RHJ-algebra.

It is then obvious that

(A/£) = (R[x,y], ?

RHJ-algebra.

W

M

n

above concerning functions

and

g

vanishes on

cannot

(A[t], i*£)

The point is, one can find an algebraic surface {(x,y,t)|x,y < 0 < t}

X(A[t],iJJ) = {(x,y,t)|x >^ 0

ijp

(A,1])11)

'£w[y])> we argue that

x-y plane to precisely the open third quadrant

is

In any case,

RHJ-algebras.

in three space in the octant

I(S) C A [ t ]

'£" = y [-y] = P[-x,-y].

In either case, we conclude simple refinements of

RHJ-algebras need not be With this same

p f = $[-x],

Let

y >_ 0}.

h(x,y)

convex. S, then

or

which projects on the

{(x,y)|x,y < 0}.

Now

But the discussion in the paragraph

non-negative on

The reason is, if f

S

X(A,'p)

implies the ideal

0 £ f(x,y,t) A(T). (This is most easily proved R geometrically, using the irreducibility of the Zariski closures S and T of

S

and

T.)

Since a homomorphism

A(S) & A(T) •> R is just a pair of R homomorphisms A(S) •> R, A(T) -*• R, it is clear that X(A(S)® A(T) ,$ (S) ®fP(T)) = R R S x T. However, in general even £(S) ® '£ (T) will be too weak an order and R will have convex prime ideals with no zeros in

S x T, as in the above example.

Contracted orders are also hard to work with, in general. if the order

p = 'p [xy - 1] C R[x,y]

the weak order.

is contracted to

R[x], one obtains

Geometrically, it is hard to reconcile

X(R[x,y] ,']$) C R ,

which lies in the half plane X(R[x],'|5 ) .

x > 0

with the entire x-axis, which is

Presumably, a contraction of an

an RHJ-algebra.

For example,

RHJ-algebra need not even be

On the other hand, 8.2.7 states that contractions are very

well behaved for semi-integral extensions. Suppose and an order where

Rn.

176

is any subset.

Then we can define

p(Y) = {f e A(Y)|f(y) ^ 0

Y = Z(I(Y))

to see that in

Y C Rn

is the Zariski closure of

X(A(Y) ,'p(Y)) = 7, where

Namely, any

all

f

Y

non-negative on

y e Y}. Y

in

A(Y) = R[X][.. .Xn]/I(Y)

Of course, I (Y) R .

= I (Y),

Also, it is easy

is the topologieal closure of Y

is also non-negative on

Y.

Y

If

Y

is semi-algebraic, then so is

Y

by Tarski-Seidenberg. n

Y C R ,

we will prove that for all closed, semi-algebraic RHJ-algebra.

semi-integral extensions A C B

f e A,

i*:

with

(B,p)

an

RHJ-algebra with

X(B,p) -* X(A, 'p n A ) .

Zariski-dense in f

closure

i*(X(B/P) C X(A/P w ).

Y = X(A(Y) ,p(Y)) -• X(A, A H']5(Y)).

is non-negative on

Pa C B

A -> B

Thus

X = X(B,'|5), then

is injective, no function

That is, i*(X(B,'£)) f 6 'j5 H A

is

if and

A

-• B a

A a = A/i" (P a ). We have

and a diagram of inclusions

i • IIB a

is the contraction of an order on n B , namely, n '£(X(B /-P ))>

QJ

f C B, but

(X

'^

OC

f

= $/P . (X

(We could also contract IItp ,

CC

OL

could be definitely weaker than

the Noether Normalization Lemma, we can obtain of some pure polynomial extension

A , and then contracting

n(A

the only point at which we might lose pure polynomial extensions

B

A [X,...X, ]

AHf by first contracting the \i\), \

of

A .

= (VV> to

Thus we can study t0

A.

form

ViY^

for some subset

that

(A , Aa n>]3(Y1))

Perhaps surprisingly,

In fact, by induction we could

RHJ-algebra order on

Y x C X(A ,']$w) x R.

was an

VXr"\]'

RHJ-algebras by contracting is in the

A -• A [X,...X, ] . K^ a <x l

a finite, real domain, and with an

By

as an integral extension

X

H 1 ^ (X ))

'JHXCB^/p )).)

attack this extension by attacking the case of one variable A

is

X(A, '£ H A ) = i*(X(B,p)).

be the minimal primes, B a = B/P a,

X(B,'P) = U X(B ,']) ) , where

then to

f CB

i*(X(B/j$)), or, equivalently, on the topological

I f CB

i:

i*(X(B/|3)) C X(A/£ ) .

IIA a

and get

is an

RHJ-algebra, under

'£ = '^(X), where

Since

inclusions of finite integral domains

since

an

X(A,'p ) . Moreover, it is obvious that

only if

The order

(A(Y),'^(Y))

A C A(Y), inducing

f ^ 0, can vanish on

Let

Y

is any extension of reduced, finite R-algebras, and

an order with consider

(A(Y),'^(Y))

So far, we have established this only (a) for closed semi-algebraic

sets and (b) for images of sets

If

Ultimately,

RHJ-algebra, with

A

-•A [X,], with A [ X ^ , of the

We would then like to conclude X(A , A^ H']5(Y1)) = T T Q ^ T ,

177

where

IT: X(A^, ^^)

R -• X(A ,^ w)

x

is projection.

However, it is not clear

how generally this holds, and even in the semi-algebraic case, we need the Tarski-Seidenberg theorem to prove it. We conclude this section by pointing out that we have established an invariant criterion that

(B,'p) G (POR)

is the affine coordinate ring of some

closed semi-algebraic set in affine space over R-algebra of finite type. Pi C B B

i

= B

with

/ p i»

R.

Namely, B

must be a reduced

Also, there must be finitely many p-convex primes

(0) = O P i , and finite refinements of the weak orders

with

^

= B n

nC-Pi) > under the inclusion

B^IIB^

^

C Bi,

(Strictly

speaking, we must still prove the basic real Nullstellensatz, 8.2.1, and 'P(X(B. /£•)) = CP-) • 1 1 p these tasks in the next three sections.)

We proceed now to

establish Stengle's theorem that

8.3. Real Curves Let

K = R(x,...x )

closed ground field

be a real field, finitely generated over our real

R, with tr. deg. D(K) = 1. K

fractions of the integral domain P C R[X 1 -..X n ] zeros in

is a prime

Thus

K

is the field of

A = R[x-,...x ] = R[X,...X ]/P, where

'^-convex ideal.

We will prove that

n

R , that is, there exist homomorphisms over

R,

R C K, with

the valuation ring of elements finite relative to place of

K,

K

A C A R , where R.

must be Archimedean closed in

K

R.

(Or, since

for any order on

can be AR C K

K

over

tr. degR(K) = 1 , K

is

The associated signed

p R , is then R-valued since a non-trivial place of

lowers the transcendence degree over

has

p: A -• R.

Actually, we will prove something stronger, namely that totally ordered non-Arehimedean over

P

R R

non-Archimedean over

R.

Thus the residue field Thus we obtain

pR:

A n /Q n is algebraic over R by results of 7.6.) K K R. K -• R, ± °° and, by restriction, p: A -• A R

Proposition 8.3.1.

Let

K = R(x1-..x )

one variable over a real closed field Let

x £ K - R, so that

the contracted order

178

K

R.

Let

is algebraic over

'$ n R(x).

be a real function field in ']) C K

be any total order.

R(x), and let

'£ C R(x)

Then (using the notation of 7.5):

denote

Case (i) a >b

If '£ = '$m [resp. 'p J , we can find

[resp. a < b ] , both orders

f

and "J5 a,—

a,+ orders on K with Case (ii)

b G R

on R(x) extend to total

A C AR.

If "jT = $c

such that for all

+

[resp. $c

] , c G R, we can find

[resp.

a G (c-e,c)], both orders

a G (c,c+e)

!P on R(x) extend to total orders on K with a, ~ Case (iii)

If f = £ D where

such that for all a G ( a 1 , a " ) , both orders total orders on K with

c = 0, by replacing

'p

v j>J .i

be the minimal polynomials for f^ . (x) G R[x].

a 1 , a" G R,

R(x), say

Suppose

y

and

m .l«y ij

i j

x.,

1 0

has real roots over

{G Q (a(x)),.. .Gk(a(x)) = h(x)}

f-signs in

in case (ii) or

R(x). We now appeal [resp.

a' < D < a"

in

(c - e, c ) , depending on whether

a G (a 1 ,a")], then the R-signs of all

agree with the

We also make our choices so that

G(y)

is an interval containing (at least)

The terms in the sequences

case (iii)] so that if 'B c,-

G Q (y),... ,G k (y).

h(x) ^ 0 G R(x).

to the results of 7.5 to choose

terms

K = R(x,y). Let

m.-l J-»i ,J

I J

{Go(j3(x)),...,Gk(j3(x)) = h(x)}

or

on R(x) extend to

••• + g m (x)

+

G Q (y) = G(y), G ^ y ) = ^

by our assumption on

one of these roots.

•IT = *B c,+

, '$

a f < D < a",

Consider the Sturm sequence for the irreducible polynomial

term is a constant

and

K

y G K so that

m. j j J J

(R(x), f).

A C AD.

x by 1/x.

0 = G(y) = g o ( x ) y m + g 1 ( x ) y m - 1

Now,

and

A C AD. K

In cases (ii) and (iii), choose

over

'J5

Case (i) occurs only if D = D R (x) = ± °° . We can deduce case (i)

from case (ii) with

G(y)

e > 0 in R

D = D R ( x ) , the cut of R defined by the

order 'p C K , is transcendental, then we can find

Proof:

such that for all

-f-signs

a(a) < 3(a)

in

of the R

G i (a(x)), G i (3(x)).

and so as to insure that

179

a

is not a root of any of the numerators or denominators of the rational

functions R(x).

a(x), 3(x) , or the coefficients of the polynomials

G.(y)

over

Finally, and this is the most important restriction, we insure that the

conditions on a guarantee that coefficients

a

is not a root of any of the leading

f n .(x) of the minimal polynomials

F.

for the elements

x.,

1 £ j 0.

transversally, but does cross

The curve does not x = a

transversally

a G (0,e).

for

3 2 In example (c) , G(x,y) = y -y -x

specializations a>0,

x = a, small

y - y - a

has three real roots over

(R(x), !pQ

a.

If

a < 0,

y -y

has one real root, and if

-a

has three real roots,

a = 0,

y

-y

has a double

root and a simple root. Example (d) has a zero at the origin, but D R (x) « 0.

total order with

Since

R(x,y)

does not support a

G(0,0) = 0, we get a homomorphism

p: A = R[x,y]/(G(x,y)) •> R, hence (x,y) C A is f -convex. However, (x,y) C A is not ('Cw ) -convex. If it were, the signed place extension d theorem of 7.7 would imply the existence of a total order on

K = R(x,y),

with signed place

Note specifically

that

2

pR:

K -»• R, ± °° extending

2

x (x-l) = y , hence

(x,y)

is not

x-1 G ('£ ) w d

Cpw) -convex.

or

p:

A •*- R.

1 < x rel ($ ) . w d

The conclusion

relative to any total orders on

x-1 £ ($w)

K = R(x,y)

we must have

Clearly, then

also shows that 1 «-«>g^ ^ K, K.

Then there exist

K ~* R, ± °° , with rank(p) = r,

p(x.),p(g.) E R,

p( g j ) > 0.

and

Proof:

Apply 8.4.1 to the real function field

E = R(x.,y.,z.) l

j

3

where

y]-zy Yfr-i. Remark.

n

An alternate proof of 8.4.3 could be given as follows.

as in 8.3.1, first any order extending one of the orders

!|5 C K 'p

+

If

r = 1,

is chosen, the replaced by an order

C R(x^) C K, by a careful choice of

a E R.

This choice can be made exactly as in the proof of 8.3.1, so as to preserve the signs of the relative to 8.4.1

R.

g.

and keep the

g.

finite, but not infinitesimally small

This technique is now combined with the inductive proof of

to give 8.4.3. Using either method of proof of 8.4.3, the conclusion can be strengthened

to include an approximation statement. defines a signed place The new order on

K

pR:

Precisely, the original order on

K -* A, ± °° , where

A

is Archimedean over

corresponds to an R-valued signed place

The conclusion of 8.4.3 can then be extended to say that the approximate the

p R (g-) E A, ± °° as closely as desired.

this just means

p(g.)

p:

K

p R (g.) = ± °° ,

can be made large.)

The next corollary constitutes a "weak" strong Nullstellensatz.

Corollary 8.4.4. g ^ P.

186

Let

P C R^.-.X ]

Then there exist zeros of

P,

be a prime n

a E R , with

'p^-convex ideal,

g(a) ^ 0.

R.

K -• R, ± °° .

p(gO ^

(If

K

Proof:

g / 0 £

R ( x r . . x n ) = K, the fraction field of

Choose a total order on

K

R [ X r . -XJ/P.

and apply 8.4.3 to the single element

g

or

-g,

whichever is positive.

Remark.

•

Corollary 8.4.4 is exactly the basic Nullstellensatz stated

previously as Proposition 8.2.1. established.

For example, if

n

{a e R |h(a) = 0 all of

a €= z(I)}.

all

Thus all the consequences of 8.2.1 are now

I CR[x i ...X n ]

h E I}, then

is any ideal, Z(I) =

*ti(I/Pw) = {f € R [ x r . . X j |f (a) = 0

If we combine this result with the characterization 2.2.4

vfa(I,pw), we conclude that the ideal of functions which vanish on

consists of exactly those

f

such that

Pi,q. E Rp^.-.X ] , h. E I.

Z(I)

S

f + Z p . = Z q . h . , for some

s _> 1,

This result, or similar versions, is known as

the Dubois-Risler Nullstellensatz in the recent literature, [14], [16], [17]. We point out that our proposition 8.4.3 as well as Corollary 8.4.4 date back to the early work of Artin, at least in essentially equivalent forms.

The

other ingredient of the Dubois-Risler Nullstellensatz is then the characterization of the radical of a hull for general partially ordered rings, given in 2.2.4. In fact, little extra work is now required to characterize those functions f E RfXj.-.X ]

which vanish on a basic closed semi-algebraic set of the form

Z(I) H w{g.} C R n . V = $„[g-] w j

is

First, assume

an order on

is an

RHJ-algebra,

f E A

which vanish on

Z(I) H W{g.}

is a radical ideal so that

and assume

J C I.

D Z(I) Hw{g,}

f

s >_ 1

...g

g X

l

X

= r

E(k2 J 0,

has no interior in '$ [g.] w j

an order on

It would then be unnecessary to actually determine

I(W{g.}).

Next we state a weaker form of 8.4.3 which is as useful for most purposes. Corollary 8.4.5 •P [gi'**gv]

a

(Artin)

A = R[x....x ]

be a real integral domain

finitely generated refinement of the weak order on

Then there is a homomorphism

Proof:

Let

Extend

p:

'^w[g^]

A -• R

with

p(g.) > 0 ,

to a total order on

A

g. ^ 0.

1 < i < k.

and apply 8.4.3.

In particular, this result applies to the polynomial ring In fact, we deduce the following, also due to Artin.

A,

•

R[X,...X ] .

(Part (b) is Hilbert's

1 7 t h problem.)

Corollary 8.4.6. (a)

g. G R[X 1 ...X n ]

If

is an order on

R[X 1...X n ]

are nonzero polynomials, 1 n X(A/}5w) Here

X(A,'B) = W

Conversely, if or

^[g^-h]

In either case, 8.4.5 guarantees U{g i> n X(A,'£w).

Thus

U{gi>

is

C Q> ) C CP ) C cp ) = (B ) =?(R n W WS Wp W m W(J In the next section we will prove a

Consider the orders R[X^...X n ].

theorem of Stengle which implies distinct orders here, f

w

C (tp ) w s

V

('£ ) = '£(R n). Thus, there are only three w P C'^(R n ). These inclusions are definitely

For example, Hilbert [ 1 ] gave an example of a strictly positive f(X,,X2), which is not a sum of squares.

polynomial in two variables, proof that

(£)

= *p(Rn)

W P polynomial belongs to

hand, by definition, term of

^[g^h]

The

X(A,'p ) .

on the polynomial ring

proper.

then either

does not vanish identically on

Zariski dense in

is an order.

U{gt> = {b G R n |g ± (b) > 0}.

P, and

and

'^1T[g.] C A W 1

P

A.

f

(p ) . w s Q3 ) w s

Thus

'B / (? ) , if n = 2. On the other w w 9 9 = {f|(l + Eh.)f = Z g . } . Immediately, the homogeneous 1 3

of lowest degree must be a sum of squares.

Hilbert's example

f(X,,X2)

But now if, say,

is made homogeneous, F(X ,X.,X 2 ), so that

f(X 1 ,X 2 ) = F(1,X 1 ,X 2 ), then hence cannot belong to

Our

will show, in fact, that any strictly positive

F(XQ,X;L,X2)

Cp ) . w

s

Thus

cannot be a sum of squares and

Cp ) w

f f-P(Rn)

if

n = 3.

s

The last result in this section is a signed place perturbation theorem for function fields.

The statement and proof are similar to 8.4.1, but

instead of trying to make elements finite relative to a discrete rank

r

189

signed place, we assume certain elements infinitesimally small relative to one signed place, then try to keep them that way relative to a discrete rank r

signed place.

This result will be the basis of our study of derived orders

in 8.6 and dimension theory in 8.10.

It provides a tool for studying the

geometry of a semi-algebraic set near a point, or more generally near a semi-algebraic subset.

Proposition 8.4.9. R

with

p':

Let

tr.deg.R(A) = r.

A = R[x,...x ]

Let

K

be the fraction field of

K ""* A, ± °° is a signed place over

p'(x^) = 0.

(Necessarily

p

1

R, with

signed place

Proof: p1

hence

p:

If

A

A

K -• R, ± °° with

and suppose

Archimedian over

= p R , relative to the order on

This follows easily from results of 7.6.) r

be a finite integral domain over

and

induced by p 1 .

K

Then there exists a discrete rank

p(x^) = 0.

r = 1, there is nothing to prove since we assume

is already discrete rank

R

1

and R-valued.

If

p'Cx^) = 0,

r >_ 2, we prove the

proposition by induction. By the Noether Normalization Lemma we can find of the

x., say

p'(u.) = 0.

u,,...,u , such that

A

r

linear combinations

is integral over

Thus we may as well assume that

A

R ^ . - . u ].

Certainly

is already integral over

R[xr..xr]. Let x

.,

f.(x

.) = Ea..(x, ...x )x

j > 0, with coefficients in

polynomials, hence applying polynomial

pf

. = 0

be the minimal polynomial for

R[x,...x ] .

we get that

Za i -(0...0)T 1 £ R[T].

Choose

The

0 £ R

0 < e G R

than the absolute value of all other roots

{a,}

f.(x

.)

is a root of the nontriviat such that

of the

L

be the algebraic extension of

j > 0, and replace on

K

defined by

namely, take

p

hence replacing rank

r

place of

190

1

A p'

extends to R C L.

K

L

by

p:

L.

is less

/e 2 - x z + . ,

y. = / e 2 - x 2 + . - e.

In fact, the place

p

1

The order

extends to

The residue field is Archimedian over

does not change our hypotheses.

K = L.

L,

R,

Also, any discrete

L -• R, ± °° restricts to a discrete rank

Thus we may assume

over

IJ

obtained by adjoining

R[x^,y.], where

= pn,

signed place K.

by

K

e

Ea..(0...OjT1

K

Let

are monic

r

signed

R.

With this assumption, given any signed place with

p(x i ) = 0 ,

1 R, ± °°.

In the notation of 8.4.9, suppose given finitely

many elements

g. ± 0, which are positive in the order on

g. G A ,

p1 :

K -*- A, ± °° .

statement that the discrete rank

r

g.

are also positive in the order induced by the

signed place

p:

is the composition of the first gj

K

Then we may add to the conclusion the

K -* 1^, ± «>->... -> K p r ^Cg^) t °» where

and moreover, we may arrange that

If

•

8.4.9 which is useful in studying

geometry near a point.

induced by

by

on infinte elements, as in 7.7.2, to take into account the

fact that the order on by

q,

R, there is a discrete rank

r-1

g.

P r j_ •

±°°->R, ± ° ° , K

~*K r y

±

°°

maps in the chain.

£ ( x r . . x n ) C A = R [ x r . . x n ] , then already

are various ways to keep the

^

p( g j ) t 0.

There

positive relative to our new order on

K.

For example, using the technique of the proof of 8.2.10, one can adjoin certain P'(g-)

/gT =e

to

K, then replace

-^R>

0 < c.

there is only a single

so that

g

{x^.-.x^,}.

R[x,...x , Jg7 - £ . ] , where

This technique works whether or not

Thus we can concentrate on the case

just try to arrange that

formula for

by

(See also the proof of Proposition 8.6.5 in a

later section for this technique.) g. €= (xj.-.x^.

A

p g.

,(g-) £ 0.

Setting

g = Ilg., we may assume

Now, we just add the dummy generator

g

to our

A = R[x,...x ,g]. Then we might as well rename the generators becomes

x^, the first element of the transcendence base

If one inspects the proof above, one finds

so that at the crucial inductive step constructing we obviously have

192

g. G (x-^.-.x^, and

p F (g) = g / 0.

pp:

g = x^ & R(x 1-. .x r _ 1 ) F(x ,y) ~* F, ± °° ,

Our extension of 8.4.9 is thus proved by

induction.

(Note that, again, if

r = 1

there is nothing to be proved.)

8.5. Characterization of Non-Negative Functions In this section we prove the theorem of Stengle [22], characterizing the functions nowhere negative on a closed semi-algebraic set. Begin with a reduced, real R-algebra

A

of finite type, and a finite

refinement of the weak order ty = p [g.]. w (Stengle).

Proposition 8.5.1 X(A,'p)

j

An element

f E A

if and only if there is an equation

(f

n

is nowhere negative on

+ p)f = q, some

n :> 0,

p.q eVProof,

The "if" statement is trivial.

We prove the "only if" statement A 1 = A[t]

by applying the Nullstellensatze of 8.2 and 8.4 to the ring polynomials in one indeterminate over Let on

A

f

= f [g.] C A'.

We know

extends to an order on

simply a homomorphism follows that

f

x:

!

X(A ,^ )

A1

A -* R

of

A. '.|5! is an order on

by 6.1.

A homomorphism

together with a value

is identified with

Af

since any order xf:

A1 -> R

x'(t) E R.

X(A/p) x R.

is

It

(This we also dis-

cussed in 8.2.) Suppose

f E A

is nonnegative on

t2+ f

on all zeros of

f E /H(t z + f/p1) C A 1 .

in

X(A',^').

X(A,'p).

Then

f €ACA'

vanishes

Thus, by the Nullstellensatz,

From 2.2, we obtain an equation

P(x,t) = Q(x,t)(t2 + f(x))

for some

n > 1,

functions on

P(x,t) E:p'.

We are regarding elements of

G(x,t) E A' can be uniquely written

G (x,t ) + tG,(x,t ) . Applying this decomposition to P(x,t), Q(x,t)

by

F(x,t2) = 0 E A' a E A.

as

X(A,'p), X(A',^'), respectively.

Now any element

any

A,A'

2

P Q (x,t ), QQ(x,t ) , respectively. implies

F(x,t) = 0 E A1

Thus we obtain a relation in

and thus

G(x,t) =

(*), we can replace But any identity F(x,a) = 0 E A,

A,

193

f 2n (x) + P Q(x, -£(x)) = 0.

(**)

Now, we investigate

P Q (x, -f(x))

more closely.

Since

P(x,t) ^ ^ ' ,

we can find a formula

P(x,t) = EPj(x,t) gj (x),

Specifically, the

gT j

gj £'j5CA.

are finite products of the

g., where 3

'p = '£ [g.] C A. w j

Thus

P Q (x,t 2 ) = (Pj )0 (x,t 2 ) + t 2 P^ 1 (x,t 2 ))g J (x)

by writing

PT(x,t) = P T

by

) +tP

-,(x,t ) , expanding, and using uniqueness 2

of the decomposition t

ft(x,t

P(x,t) = P Q (x,t ) + tP,(x,t ) .

Replacing

t

Thus (**)

f n (x) + p(x) = q(x)f(x)

gives an equation

We obtain 8.5.1 by multiplying this last by

CoroHary 8.5.2. '-P = '-P [g • ] w j

a

and

f)g(x)) j

p(x),q(x) G'^.

f(x).

•

be a reduced R-algebra of finite type,

Suppose

g GA

A,

vanishes on

f £ A

a function

Z(f) H X(A,f) , so 0 £ g2n ).

g G v^H({f}, $) C A.

for some

A

Let

with

finite refinement of the weak order on

nowhere negative on

1

t

- f(x), we find

PQ ((x, -f(x)) = (Z P x f)g(x)) -f(x)(E f(x)(E Pj(# 1x Pj() 0 (x, -f)gj(x))

that

by

This is a consequence of the proof of 8.5.1. again, and

'£' = ? w [g k l

c

1

A .

Then

We introduce

g e >^M({t^ + f}, f1) C A 1 .

Going through the proof of 8.5.1 8.5.] yields an equation

g

+ p = qf,

p,q

which is exactly our assertion.

Remark. the hypothesis

Corollary 8.5.2 should be compared with 2.2.3. f £'|5

of 2.2.3 by the weaker hypothesis

f

We have replaced non-negative on

X(A,']>), and have established the same conclusion in both 8.5.2 and 2.2.3.

194

Corollary 8.5.3. '£

=

a

'P [§•] w 3

function

Let

A

be a reduced R-algebra of finite type,

finite refinement of the weak order on X(A/p).

strictly positive on

1+p = (l+q)f, some

Proof. setting

a

p,q ^='p.

But now

p'

(l+q')f = p f ,

is also strictly positive on f

8.5.2 again gives

f €= A

Then there is an equation

Corollary 8.5.2 gives equations

g = 1.

A, and

1+p" = q"p ,

p",q" £'p.

p! ,q' e 'P X(A,'.p).

by

Applying

1

Now check that

(1+q )(l+q")f =

1 +p' +p".

D

Note that the equation e

CA

*s(i)

Let

1+p = (l+q)f

A

be a reduced, real R-algebra of finite type, fp = 'p [g.] w j A.

is an RHJ-algebra and that the functions coincide with

the orders

consists of all functions

(Y - X 3 )

(p/I)

A

I

C A/I

n >_ 1, is

p,q G ])}.

A = R[X,Y],

wil1

2

f G A

I = (Y - X ) , then

R 2.

congruent to

(Consider

3f/3X

f £ (*p /I)

then

(l+p)f = q, some

t

f £ A

P

er

is nowhere X

modulo

at the origin.)

('P /I)

is easily seen to be non-

C (p/I)

by Stengle's theorem.

is a reduced real R-algebra of finite type and

the ring of functions $ = 'Pw[t5]>

X

P

r0

= CP/I)p> since, first, 'P/I ° P / I , which gives

2(1) C x, which gives

with

be a

3

C A/I

On the other

of the functions

a finite refinement of the weak order, then the order all

By 8.5.1, fly I)

Z(I) C X.

Z(I)

f (X,Y)

Suppose

!p -convex by 8.2.2, and

In general, ']) /I C CP/I) p

X.

(A,'.p)

X = X(A,'p)

1

are defined.

Z(I) C R , but no polynomial

Cp/I)

nonnegative on

non-negative on

Cp /I) , and, secondly, any

negative on If

Then

can be non-negative on all of

We do have c

f G A/I

For example, if

negative on

f £ A

consists of the restrictions to

non-negative on all of inclusion.

CP /I)

a

We have now established that

+p)f = q, some

^-convex ideal.

Cp/I) , 'p / I ,

'p /I

n

•£ = {f G A|(f

is a radical

hand,

f = 1+p/l+q

s(i)-

finite refinement of the weak order on

I CA

can be written

A

'$ = 'p [g.]

is

'P C A, consisting of

p,q G'p, is not a geometric invariant of

on the set

X(A,'p).

For example, if

A = R[t] ,

^^s"

On the other hand, if

f

is strictly positive on

X(A,'p), then 8.5.3

195

guarantees that

f G'j5 .

contraction to

A

Also, recall that

of the order

'pcfl. C A

characterized as the ring of functions on all

f G A

with no zeros on

X.

C f n

.

Now, A c f 1 . can be invariantly

X = X(A,'p)

f

pG'P,

is perhaps a more natural geometric invariant of '£ by

£ ( X ) , then

$(X)cn-\

(A(X) c r i ., $ ( X ) c r i O

than the ring A

X

( )CM^ X.

is

Ag^.

A.

If

invariantly

Recall from Chapter V

Stengle's theorem has a nice interpretation

in terms of the structure sheaf associated to X

The ring

is the ring of global section of the structure

sheaf associated to (A(X), '^(X)).

nowhere negative on

X, then

g G A.

X c

characterized as the functions nowhere negative on that

obtained by inverting

is nowhere zero on

and by 2.2.4, 1+p = fg, some

we replace the order

is defined as the

This is immediate from 8.5.3, or directly

from the Nullstellensatz, since if 1 G H({f}, p)

'p C A

if and only if

(A/p).

Namely, f G A

is

f G ' P g ^ C Ag, f . . That is, f

be "positive" in the ring of sections over the basic open set

should

D(f) C X.

Stengle's theorem also generalizes Artin's result 8.4.6(bJ, in the case A = R[X X .. .Xn] , p = £ w . nowhere negative on Eh^, some

h^}.

Namely, it is trivial from 8.5.1 that if

n

R , then

$ ww)) . f G ($

f

is

2

($ w : f) = {h G R[X 1.. .XR] | h 2 f =

Let

It would perhaps be interesting to characterize the zeros

Z(pw : f) = Z(/H(Cpw: f) , ?J) C Rn.

By 2.2.3, if ZCPw : f) = 0 , then

and conversely.

By Stengle's theorem

Z('^ ' f) C Z(f).

s t include all zeros

x

Z($ w : f)

that

mu

of

of lowest degree in the Taylor expansion of

f f

f G Cp w ) g ,

It is also easy to see

such that the homogeneous term about

x

is not a sum of squares.

8.6. Derived Orders Let

A

be a reduced, real R-algebra of finite type, 'J5 = '-P.Jg-] w j

finite refinement of the weak order have identified

X

X = X(A,'^3).

some

n >_ 1, p,q ۥ]>}.

Also, if orders

196

I C A

In previous sections we

with the maximal convex ideals of

that the functions nowhere negative on

X

(A,'jJ) and established

coincide with

']3 = {f G A| (f n +p)f = q ,

In this section we study the derived order

is a radical

a

'^^

of 3,12.

p-convex ideal, we investigate briefly the

(1> / I ) d and, when defined, the orders

'^/I

and

C^d/I) •

All of

these orders are geometric invariants of the rings of functions X(A,p),

p,q G p

and

p

CP/I)^-

*4>,/1, when defined.

zation of

']3^ consists of those

not a zero divisor.

include, of course, the orders

on

Our study of

An equation

pf = q,

nowhere negative on where

f

with

pf = q,

'£,, ^ = ^^[g.]

(Hp.), = nCPO

Note that since

! d

we will also have described

U(-f) C x

f GA

will

Also, we will then have a characteri-

Q

f

A/I

X(A/I, 3 V I ) , respectively.

Recall that the derived order some

A,

by 3.12,

! d

'£, for a finite intersection of orders of type

p,q £'.p,

p

X = X(A,'|5).

is negative.

not a zero divisor, does not imply

The function

p

could vanish on the set

We need to distinguish between "degenerate"

and "non-degenerate" points of our semi-algebraic sets. In general, let

(B/p)

is degenerate if for some and

be any RHJ-algebra, f,h G B,

y e U(f).

h(y) = 0, all

h

f

points of

X,.

X

by

with

(We will write

X = X(B,'^).

We say

not a zero divisor, we have h(U(f)) = 0.)

definition of degenerate point depends only on particular order

X = X(B/P).

X

and

x G X x G U(f)

Note that the

B, not on the

We denote the set of non-degenerate

The subscript refers to "derived" or "dense" for

reasons which will appear shortly. Given any point as follows. U C U' I

=

For each open

implies U

I(U)

h i (U(f i )) = 0. hence

I

U(f) C X If I in

x £ X

we can associate an ideal of degeneracy U,

x £ U, let

f

I(U ) C l ( U ) .

Since

I(U) = {h G B|h(U) = 0}.

Now, if

U(f) C U{f.}

^,...,11,, with

is chosen by 8.1.1, then

Choosing an affine embedding

as a small open ball in

R n , intersected with

is an integral domain, it is clear that

For general

the minimal primes.

B,

x G Xd

if and only if

This is equivalent to

I

Recall that the

order

CP.

We claim that

I x

I A = ( 0 ) C B/P . J

x £ X,

h.^ e I(U(f)),

x

if and only if is Zariski dense

I x C U p . , where

P-

if and only if

J

From 8 . 2 , Z ( P . ) = X ( B / P , , tp/p ) J J J

x ^ U ^ ) ,

X.

C P . ( some

argument in commutative algebra. '$ C B.

Clearly

X C R n , we can interpret

= (0), which just says that any open neighborhood of X.

C B

is Noetherian, the ideal

is finitely generated, say by

= I(U(f)).

B

B

I

P. C B

are

j , by a well-known

are convex for any x e Z(P.)

and

J

and

X(B/P) = U X ( B / P . , p / P . ) i

197

The "if" part of the claim is easy, since if some a neighborhood

U(f) of

x G X, then

U(f) H Z ( P . ) , contradicting 3

on

I C p., x J U(f)

f G p., 3

since

f( x) > 0.

hf = 0 G B.

h G B/P.

Thus

x G Z(P.)-

vanished on

is non-zero, but vanishes

I = (0) C B / P . . x 3

Then any element

h £ P.

Conversely,

h G n ^

P. - p. i 3

Finally, if

suppose vanishes on

hj £ P.

vanished

on a neighborhood

U(f) n Z(P-), choose h 9 G O p. - p and consider 3 I ± j . i 3 We have h £ P., h(Z(P.,)) = 0 , i / h, hence h(U(f)) = 0,

h = h1h2. since

U(f) = U (Z(P.) n u ( f ) ) . x i

prove

the following.

X(B,1>) , =

Proposition 8.6.1.

Proposition 8.6.2, (0) = n p

with where

U X(B/P.,?/P.) . d P i minimal

Suppose

and suppose

B -•IIB.

The arguments in these last two paragraphs

p. C B

is a finite collection of primes

'^ C B i = B/P^ are orders.

is the natural inclusion, then

1

Proof: U X(B., p . ) . ^ l l

x G X(B, '^) , a

If

x G X(B., '^.) , then 3 3 d

but

x £ X(B., 15.), let

X(B,'.|>) , = U d Pj minimal

U(-f) .

if and only if

f G B,

f(x) < 0,

Thus

U m

3 d

I C P., some minimal x 3

and, again, any

The proof that

just like the last step of the proof of 8.6.1, with the the

3

If

P.. 3

I C P., x 3

f (mod P.) G -jj. C B., then

X(B., '^.) i i

x G X(B., f.) . J j

X(B., ']>.) .

X(B, %]>) =

I C P., just as in the proof of 8.6.1. ^ J

xGU(-f)CX(B,f)-X(B.J.)C 3 3 vanishes on

If '£ = B nl!']).,

The starting point is 8.2.9, which asserts that We know

D

hG

n P. -P. . ^ 1 3

x G X(B., '^.) 3 3d

is

X(B., 'p.) replacing

Z(P i ).

•

Proposition 8.6.1 seems more natural than 8.6.2, since it applies to any RHJ-algebra.

But 8.6.2 is better since it applies more directly to the

situation of 8.2.11, where we showed how any closed semi-algebraic set is of the form

X(B, •]>) .

We next show how non-degenerate points behave with respect to intersections of orders.

Proposition 8.6.3. i £ i £ k, and suppose

198

Suppose

P C B

(B, 'JK)

are RHJ-algebras,

is a '^-convex prime ideal.

']$ = n-j^,

Let ]3. ,

denote

those

p.

'p..)

Then

1

so that

P

is

X(B/P, $/P)

Proof:

a

'£., convex.

(Thus if

P = (0), we get all the

= U X(B/P, ?.,/P) . i1 -1 d

First note that by 2.7 we know some

'p.,

exist.

If

x G X(B/P, .p. ,/P) , but some f G B, f £ P, vanished on a neighborhood U 1 d of x in X(B/P, $/P), then f also vanishes on U nx(B/P, f.,/P). This contradiction shows

U X(B/P, *.p.,/P) C X(B/P, 'P/P) . 1 d d i1

Conversely, if

Let P

f. , £ P is not

vanish on a neighborhood of

'£. ,,-convex, where the

follows that f.,, 4 P

P

x,

x £ X(B/P, p.,/P) 1 d

vanish identically on

Since

'.p.,, are the other orders among the H(P, 'p.M) C B.

if.

all

U(h. ,) C\ X(B/P, 'p.,/P).

is properly contained in the hull

From 8.1.1, choose f

x G X(B/P, 'p/P) , assume d

'p., it

Thus let

Z(P) n X(B, '£.„) = X(B/P, p/P) n X(B, 'p.,,).

x G U(h) G n U(h i f )

vanishes on the neighborhood

U(h)

and let of

x

f = II f±1- II f i M £ P.

in

X(B/P, '.p/P)

Then

since

u(h) = u [u(h) n x(B, 'p.,) n z(P)] U U [u(h) n X(B, £ in ) n z(P)] . i1

x

Remark 8.6.4.

•

i"

The study of derived orders of the type we are interested

in has already been reduced to the case of integral domains in Proposition 3.12.1. Thus, if

B

is a ring,

P. C B

a finite collection of primes with

and we have either

(1) an order

'p. C B. = B/P.

$ = B nil p., then

i

l

l

'P, = B n n('p.) d x

and

f C B l

with all

P.

convex, or

'P, = B O IlCp/P.) a

i d

(0) = n P^ (2) orders

in case (1) and

in case (2).

d

The next result is the central result of this section, and gives the relation between non-degenerate points and derived orders in a crucial special case.

Proposition 8.6.5. domain over on

A,

R.

g i t 0.

Moreover,

Let

Let

']3 = 'Pw[g-]

Then

(A, £ d )

p , = { f G A| f (x) > 0 a

A

•

—

be a real, finitely generated integral be a finite refinement of the weak order

is an RHJ-algebra, with all

x G X(A, '£),}.

coincides with the following subsets of

a

X(A, 'Pd) = X(A, '£) d .

Finally, X(A, 'p), also a

X = X(A, 'P w ), (the irreducible real

algebraic variety associated to A ) : 199

(i) {x G X |exists total orders p D p

(ii) {x G X|for all

g e A

with

on

A

g(x) > 0,

with

!p[g]

x

'^-convex}

is an order on A}.

Before giving the proof, we discuss some applications. the Proposition gives us a class of RHJ-algebras not at all obvious that

Corollary 8.6.6. 'P = V [g-1 Vd

a

If

B

for which it is

is a closed semi-algebraic set.

is any reduced real R-algebra of finite type,

finite refinement of the weak order, X = X(B/p), then

= {f G B|f(x) _> 0

Proof:

X(A/pj)

(A/p^)

Let

all

x G X d >.

P. C B

be the minimal primes.

and by Remark 8 . 6 . 4 , ' p , = B n i I C P / P . ) . a 3d

But

By 8.6.1, X d =UX(B/P., 'P/P.) ,

' p / P . = 'P [Wg . | 1g . £1 P . ] , and by J J

8.6.5, f G CJ5/P.) if and only if f is non-negative on J d Thus f G *.p if and only if f is non-negative on X,.

Next, let with

S

S i = Z{f ik >

in n-variables. primes above

n

W{g ik > C R , as in 8.1, where

Then

= R

I(S.)«

i a ia We also have the order p.

P.

FrOm

S 2Al

-

S = X(A(S), 'p), where

among the

we have

>

the

occur.

are polynomials are the minimal

= '$ [g., ^ P. ] W

IK.

delusion

p = A(S) nil']). .

On each ring

A

P

on

XCt

A(S) -• n A i a

and

There may be repetition

P. , but a given prime occurs at most once among the

We can collect similar terms, and denote by

S = U S^ ,

ia

ICt

[ x i " 'XJ/Pia'

we have

•

f i k , g.fc

I(S) =Hi(s.) = f l O p . , where

1

ia

X(B/P., '-P/P.) . J J d

be any closed semi-algebraic set, and represent n

1

A

First note that

P. , fixed i.

the various primes which

= [R[X,...X ]/P , put the order

'P

= O'p. , the

intersection taken over those indices with A(S) -• IlA

and

'P = A(S) H n -p .

ot

P. = P . We still have inclusion ia a Denote by P o the minimal primes of A(S).

ot

Necessarily, they all occur among the

Moreover,

p

P . a

$o = H ' p . o , so from 3.12, CPJ |i

ip

From Remark 8.6.4, "p, = A(S) nn(']5o)d a P

= n C-P-o) • 1

p d

refinement of the weak order on the integral domain

negative on

X(A O , 'P1 • o) . P P d

iK

200

ik

Q]• p

p. o 1

is a finite

P

A o , namely, 'p. o = P

'P [g-vlg-i ^

P

w

Now

Each

d

P

8.6.5 describes

CP-Q)

ip ^

!p

as the functions in

Finally, from 8.6.2,

a

S, = X(A(S), 'P) , = d

A

p

nowhere

U X(A O , *pj . From 8.6.3 (with Pg minimal P P d U Pg minimal

P = (0)), we conclude

U X(A O , p. ) , and moreover, pd , C A(S) P ^ = Pg P XP d

of functions

S, a =

consists precisely

f nowhere negative on S,. We summarize these arguments in

the following.

Corollary 8.6.7.

Let S be a closed semi-algebraic set, 'p C A(S) a

specific order of the form considered in 8.2.11 with

S = X(A(S), 'p). Then

for any order p' with p C p ' C'P(S), we have that

(A(S), 'Pp

is an

RHJ-algebra and S d = X(A(S), p ^ ) .

Proof: 1

p ,

The argument above gives the result for p' = p. For any such

p d C ^ Cp(S)d-

on

But if f eip(S) d , trivially

f is nowhere negative

S d . Thus ? ( S ) d Ctp(S d ) =1> d .

We remark that if '£" C $

•

is a weaker order with

S = X(A(S), '£"), then

it is not clear when '£" = 'p(Sd ). In fact, we do not assert that if (A,'p) is any RHJ-algebra, then necessarily

(A,'p,)

is also an RHJ-algebra. Our

arguments definitely use properties of orders of specific types. Next let (A/p) be an RHJ-algebra with '$ = '^^[g^ of the weak order.

a finite refinement

Let I C A be a 'p-convex, radical ideal

X = X(A,'p).

Corollary 8.6.8 (a) I is '^j-convex if and only if each minimal prime I

is p d -convex if and only if P^^ = I(Z(Pi) n x d )

P. C A

over

if and only if

I = I(Z(I) H x d ) . (b) If I is £ d -convex, then X(A/I, *Pd/I) = Z(I) to

Z(I) ^ Xj

n

Xd -

Moreover, ^ d / I C A/I

of functions

X(A/I, Cp/I)d) = X(A/I, 'P/I) d-

functions

is an RHJ-algebra with

consists of the restrictions

f ^ A nowhere negative on X-..

(c) For any radical 'p-convex with

(A/I, $ d /I)

I C A,

(A/I, CP/I)d)

Moreover, CP/I) d

is an RHJ-algebra,

consists of the

f ^ A/I nowhere negative on X(A/I, $/I)j.

(d) Op ) = 'pd C A, hence for any 'p-convex radical

I C A,

CP/i)d = CPp/i)d = C^/i) p ) d . 201

Proof:

(a), (b), (c) are routine restatements of various results.

first statement in (d) holds, since by 8.5.1 and 8.6.5 CPd)

have

Tne

= '-Pd-

last

P

art o f

d

( ) holds because CP/I) d

finite refinement of the weak order, so p p /I C CP/I) , so we have

c

Q}/I) d C CP p /I) d

=

and we

always

is still a

Also, by 8.5.1,

(CP/I)p) •

•

c

(1*d/I)

A/I, when

I C A

'$£-convex. We state the following without proof.

(A/I, CPd/I) )

Proposition 8.6.9. (Z(I) H X,) .

Moreover, f £ (;£,/I)

is an RHJ-algebra, with

if and only if

f

I = P C A

too hard. 8.6.5.

a prime

*£d-convex ideal.

X(A/I, CPd/I) ) =

is non-negative on

Using techniques above, one can reduce to the case and

Pd>

£/I C A/I

(CP/I) ) •

An interesting order not covered by 8.6.8 is is

C:

£

The

A

an integral domain

The last two assertions are not

But the RHJ-property seems to be a fairly strenuous extension of

We will provide the necessary ingredient for this extension in 8.10.

We now return to the proof of Proposition 8.6.5.

We follow the notation

in the statement of that proposition.

Proof of 8.6.5: we must have

f

First, if

non-negative on

neighborhood of a point of Thus, points of

X(A, '$),

are exactly the points of

-p on

A.

h(U(f)) = 0.

If Then

x

p G £,

Otherwise, h

h ^ O , then

would vanish on a

X(A,'$),, contradicting the definition of are

$d-convex. x

h f £ 0

on

By 7.7.3, the '^-convex points

f ,h G A ,

X(A, $ ) .

h j- 0,

By 8.4.3, f

'£, that is, -f

We have now proved

X(A, *#)J-

is convex for some total order refinement

x £ X(A, ']5)d, choose 2

h 2 f = p,

say

X(A,*.J3) ,.

for which

in any total refinement of Pd -convex.

f G •£

e>

^d-

Since

f (x) > 0

and

is therefore negative -f(x) < 0,

x

is not

X(A,v£d) = X(A,'J))d> and have established the

characterization (i) of this set. Next, suppose find a signed place Here, K

f £#d. p:

Then

'£[-f]

is

K -• R, ± °° , with

is the fraction field of

A.

an order on

By 8.4.3, we

p(A) C R , p(g i ) > 0,

But then

which is convex for the total order refinement of

202

A.

p

gives a point '£ defined by

p(-f) > 0. x £ X(A,'p) p.

Thus

x

is

'^-convex and

f(x) < 0.

This proves that

x e x ( A / P ) d } , since

X(A/P) d = X(A,$ d ).

Third, if

and

x ^ X(A,'£)d

orders for which we see that whenever

x

£[g]

'^[^1

is convex.

A

In particular, U{g^,g}

g

is positive in all total

Since some such order exists, refining •£, A.

is an order

x £ X(A,'|5)d, we can find

an order.

g(x) > 0, then

is an order on

c

p d = {f G A|f(x) _> 0, all

Conversely, note that by Remark 8.4.7,

U{f i> C X(A,'.pw)

h,g e A,

h ? 0,

is Zariski dense.

g(x) > 0, and

is not Zariski dense in

X(A/Pw)

This establishes characterization (ii) of

Finally, we come to the hard part of the theorem. P C A with

is a

open, hence

of

X(A,'^)

is closed.

is not

X(A,']3)... We must show that if x G Z(P) n X(A,'£)d, X(A,'£)

is obviously

X(A,'^) ,, which will imply by Tarski-Seidenberg that

contains functions in

A

(This is not clear yet since the definition as quantified variables.

finitely many distinct ideals of degeneracy we could conclude that

I

C A

If we knew only x G X(A,1^),

occur, for

X(A/.p) , is semi-algebraic without the new characterization.)

But even granting all this, we would still only know that semi-algebraic.

*])[g]

In a later section we will give another

is a semi-algebraic subset.

X(A,p),

hence

Note that the set of degenerate points of

characterization of X(A,'.J3)j

h(X(A/P) n U(g)) = 0.

'^-convex prime ideal, g £ P, then there exists

g(x) ? 0.

Now if

X(A,'p)d

was closed,

If unproved Proposition 8.1.2 were available, then we would

have a proof that

is an

(A>Pd)

RHJ-algebra by the constructions in 8.2.

However, we must make do without 8.1.2 and give a direct proof of this part of 8.6.5. Assuming '£, with a domain

P

P C A

is

']5 convex, we can find a total order

still convex, by 7.7.3. B, integral over

By adjoining

/gT

to

A, with a total order extending

'^' on

A, refining

A, we can construct '£' on

A.

(We

use the method described in the proof of 8.2.10 and adjoin no more of the than we need to insure that the weak order on

B

contracts to

By the going up theorem 6.4.2, we can find a convex prime order on

B, lying over

any total order on Let that on

K B.

B

P C A.

The reason for passing to

will refine

'P [g.]

be the fraction field of Write

on

'-P = '-P [g«]

Q C B B

/gT on A.)

for this total

is because now

A.

B, with the total order induced by

B = R[x,...x ] , and suppose

x. = x. (mod Q ) , 1 £ i

*• K

is the composition

chain of prime ideals

r

refining p:

(0) C p

-

»

'£, all finite on

K ~* K , ± °° .

C • • • C p

A, with

P = kernel(p| ) ,

In particular, we obtain a

= P C A , all convex for a total

$.

D

8.7. A Preliminary Inverse Function Theorem In order to make sense of differential topology over an arbitrary real closed field, it is imperative to investigate purely algebraic versions of the inverse function theorem.

The result proved in this section is a rather

special case of a better algebraic inverse function theorem, but is strong enough to provide a good picture of a real algebraic variety near an algebraic simple point.

This application will be given in the next section.

This, in

turn, will be used to stratify arbitrary closed, semi-algebraic sets and prove that any such is the maximal convex ideal spectrum of an RHJ-algebra.

Also

crucial for this discussion will be the work on derived orders in the previous section.

(We refer again to 8.1 for the distinction between closed, semi-

algebraic sets and closed semi-algebraic sets.) Before stating the main theorem of this section, we digress a bit in order to put in perspective a consequence of the Tarski-Seidenberg theorem which we seem to require at this point. is a polynomial and n

The result is that if n

B = {(x....x^ € R |a. £ x^ R.

y:

As discussed in 8.1, the Tarski-Seidenberg theorem easily can be used to prove certain sets are semi-algebraic, but it does not give much information on whether sets are closed or open, without further work.

On the other hand,

our going-up theorem for semi-integral extensions provides a nice tool for concluding that certain sets are closed. 8.2.7, where we showed that if

A C B

This was formalized in Proposition

is a semi-integral extension of finitely p C B, then image (X(B,'p)) =

generated R-algebras relative to an RHJ-order X(A,£ H A ) , under the projection is always closed. image(X(B,'P))

If

X(B,'£)

X(B,$W) -+X(A/£ W ).

Of course, X(A/P H A )

is semi-algebraic, Tarski-Seidenberg implies

is also semi-algebraic.

As an example, we can deduce that a polynomial

f(x,...x )

maximum value on any bounded closed semi-algebraic set the graph of

f

over

S, say

F C R

closed semi-algebraic set and the Moreover, (A(F),$(F)) We project

F

n

x R = R

n+

RHJ-algebra

is semi-integral over

.

Then

n

S C R . F

(A(F)/£(F))

assumes a Consider

is a bounded has

R, hence also over

X(A(F),^(F)) = R[y] C A ( F ) .

onto the y-axis and get a bounded closed, (comma) semi-algebraic

subset of the line.

But in dimension one, the distinction between closed,

semi-algebraic and closed semi-algebraic obviously is unnecessary, both notions simply corresponding to finitely many closed intervals (including single points and closed rays).

Thus, bounded, closed, semi-algebraic

implies a maximum element in dimension one. Now, as another way of applying the Taski-Seidenberg theorem, one could draw the same conclusion about maximum values directly from the fact that it is true in the case of the real numbers.

But this is a "transcendental proof",

whereas we have just given a "purely algebraic proof". We now state an inverse function theorem.

Proposition 8.7.1.

Suppose

Y ...Y

£R[X,...X ] .

We regard the

207

Rn

as functions on R

n

n

-• R .

Assume

and we regard the n-tuple

Y(0) = 0

B(0,e) C R n

ball

(b) Given

as a ma

e > 0

of radius

Then:

such that the map

Y

restricted to the closed

e at the origin is injective.

e > 0, there exists

6 > 0

B(0,

n

— (d) Let

A(x) = ((3Y./3X.)(x)), the derivative of

Y

Then sufficiently near the origin the inverse function tiable with derivative

A

, in the sense that if

Proof:

x e Rn.

X = X(y)

is differen-

y = Yfx ). oo

X(y) - X ( y 0 ) - A " 1 ( x 0 ) ( y - y 0 ) H X ~ X0 H

lim

y -• V

at

=

The point to be made at the outset is that once we have the maximum

value property of polynomials on closed bounded semi-algebraic sets (which we have just established purely algebraically) one can write out word for word one o f the standard proofs of the inverse function theorem for real numbers entirely in elementary algebraic terms. Part ( a ) , the local injectivity of Y necessary to find

e > 0 and a constant

is very easy since it is simply c > 0

such that for all x,x f £ B(O,e),

we have

Hx-x'll -| , hence P(x)

If

2 (Y. (x) - y.) x 1 i=l

Directly from (*),

Let us now assume x G B(0,£ 2 ).

is non-singular for all

the function

B(0,e 2 )-

-|- < ||Y(x)ll.

Y.

does not assume a minimum value on the boundary of B(0,e2)

is an

interior point at which

P(x Q )

is

minimum, then

0 = (3P/3Xj)(xo) = E2(Y i (x 0 ) - y ^ ( O Y . / S X ^ (xQ))

for all

1 < j v y y o X(y)

A 0 (x-x 0 ) - (Y(x) - Y ( x 0 ) ) f x - x o | = || x - x o || I y-yo || "

and differentiability of

Y(x).

We discuss further the differentiability of the inverse function

X ( y y lim — — £-0

+£,...y) - X.(y —

D

X = X(y).

Once we have (d), it is easy to see that the partial derivatives

(3X./3Y.)(y ) = J X °

'

y ) —

209

exist and, in fact, that the matrix yQ = Y(x Q ).

where

((3X./3Y.)(yQ))

Thus, Cramer's rule expresses the

coincides with (8X./3Y.)(y )

det(A(x ) ) .

Continuing this discussion, it can be shown that the infinitely differentiable functions of x Q = X(y Q )

computable in terms of

X. - x . , where

x

y.

X. - x.

= (y,...y ) .

Y^-y^

(il) ,..*..(y

Alternatively, the total differential

dY

as

The constant

((3Y i /8X.)(X Q ))(X-x Q ).

as power series in the

CY1-y1)V...C Y n- yn) « will then be

dY:

The higher derivatives are formally

= (x.....x ) , y

i

function

are actually

Y. - y. = Y.(X,...X )-Y.(x.....x )

terms vanish and the linear terms will be solve for the

X.(y)

as follows.

First, rewrite the polynomials polynomials in

as

(8Y i /8X.)(x Q ),

specific rational algebraic functions of the polynomials with denominator

A(x ) ~ ,

Then formally

The coefficient of

, O^/BY 1 ) (yp) , I = Ci r ..!„).

can be regarded as a polynomial

R n x R n -*• R n x R n , linear on the second factor at each point of

the first factor, and, moreover, dY has non-singular differential at (0,0). 2 The second derivatives (3 Y./9X.8X,) then occur as part of the first derivative i 3 K of

dY.

Applying the general discussion of first derivatives to

the inverse function of of the

Y

dY

shows that

is twice differentable, and leads to a computation

(3 2 X./8Y-3Y V ). 1

J

K

Here is perhaps a more algebraic approach to the derivatives The derivation to a derivation

Z/dY±: D±:

R[Y1...Yji] ->R(Y 1 ...Y n ) CR(X 1 ...X n )

R [ X r . .XR] -» R(Xj .. .X n) .

a o (Y)X m + a 1 (Y)X m " 1 + ... +a m (Y) = 0

(3X./3Y.).

extends uniquely

Specifically, if

is the minimal polynomial for

f.(Y,X.) = X.

over

R[Y,...Y ] , then we must have 0 = D i (f j (Y,X j ))

= ( O a o / 8 Y . ) X m + . . . +(3a m /3Y i )) + ( O f ./3X.) (Y,X.))Di(Xj) ,

and this last equation can be solved for

D.(X-).

Using the inverse function theorem 8.7.1, we can easily establish the following implicit function theorems.

Proposition 8.7.2.

210

Suppose

Yj ,... ,Y R e R[XX .. . X j ,

k < n,

Y^O) = 0 ,

and suppose the vectors

d Y ^ O ) = ( ( B Y ^ B x p (0),.. ., ( S Y ^ B X ^ (0)) ,

are linearly independent.

Reordering the variables if necessary, assume that

{dY.(O), 1 < i < k; dX.(O), k < j < n}

i

—

—

3

0 G R k , the equations

Then

F

Define

Y^x) = c ^

is locally

F: 1-1

span

—

uniquely define the coordinates

Proof:

1 < i < k,

1 < i < k,

x-^.-.x^

Rn

Rn.

by

Then, sufficiently near

c± G

x-(x 1 ...x k ,x k + 1 ...x n )

as functions of

x

ic+]/--xn

F(x) = ^(x),...,Y k (x),X R + 1 (x),...,X n (x)).

and onto by 8.7.1.

The result then follows easily.

n-k

Surfaces

Remark.

i i i k-

Surfaces

Y..(x) = c±,

D

Proposition 8.7.1 and the discussion of derivatives above also

gives the tangent plane of the surfaces (8Yi/8X.)(O) = 0 ,

Y i (x) = c^.

j > k, then the tangent plane to

at the origin is the coordinate plane general, the tangent plane of

x =

(^

Y^x) = 0

For example, if Y i (x) = 0 ,

••• = x ^ = 0 } = R

1 contains degenerate points which are algebraic simple points of g >. 0

Z(P)

Figure (a)

Figure (b)

X(A,$ ) . We will define the algebraic simple points of

X(A,p)

to be the

non-degenerate algebraic simple points, that is, simple points of belonging to

X(A,'£,).

X(A,£ )

Propositions 8.8.1 and 8.6.5 show that these simple

213

points are, in fact, dense in X(A,£ d ). and

g.(x) > 0, then

'£[g-]

Namely, if x G X(A,£ d) = X(A,£) d

is an order on A and 8.8.1 guarantees simple

points exist in the neighborhood

x G U{g-} C X(A,'£),.

Now, however, we should check that an algebraic simple point of X(A,'p ) is automatically non-degenerate.

This requires a somewhat careful argument.

In fact, we will base the proof on the inverse function theorem.

The following

is the central result of this section.

x G R n , with

Let g,f ,... ,f r G R[X] ,

Proposition 8.8.2.

f^x) = 0

df i (x) linearly independent, g(x) > 0, and assume the neighborhood x

is sufficiently small.

In fact, P Moreover,

Then the ideal

I(Z{fi>

n

U(g)) = P

is the unique minimal prime of the ideal P

is convex for any order

is prime.

(^...f ) , with

£ = p [g.] C R[X], with

at

U(g) of x

vanishing of the

f., 1 < i < r. l — —

A = R[X]/P

in the variety

Z{f^}

g. are positive in the order

for

•£ [g-]. This proves the last part of the proposition, w 3

If A

Proof:

A = R[X1...X ]/P.

that

P

df.(x)

zeros.

f^ near

'•P

'P [g-]

214

x and P

a

If A

R and

f 1 ...f r G P,

Rank considerations show

The zeros of P near

Thus every neighborhood of x

Corollary 8.8.4. =

By assumption, there exists

linearly independent.

we are in the situation of 8.8.2.

is convex

x G x^.

is necessarily a minimal prime of (^...f ) , and since

the zeros of the

P

is a real finite integral domain over

is a simple point, then

r = codim(P), with

U(g) C u { g . } . l

$(Z{f.} fiu(g)) C A, hence

x G X = X(A,'J)w)

Write

is a ring of

defined by the

If g-(x) > 0, then choose 3

The

Corollary 8.8.3.

In practice,

can be thought of as small balls centered

x. The first part of the proposition says that

functions on any neighborhood of x

x G Z(P).

g.(x) > 0.

Before indicating the proof of 8.8.2, we give applications. the "small" neighborhoods

U(g) of

x G Z(P),

x thus coincide with

is exactly the ideal vanishing on these is Zariski dense in X, as desired. •

is a real finite integral domain over R,

finite refinement of the weak order on A, g. ^ 0, then the

non-degenerate set simple points

Proof:

X(A,'£),

x £ X(A,£ )

is exactly the closure of the set of algebraic with

g i (x) > 0.

We know from 8.8.1 and 8.6.5 that any neighborhood of a non-

degenerate point contains such simple points.

Conversely, 8.8.2 guarantees that

a function which vanishes on a neighborhood of such a simple point is already 0

in

A.

•

Corollary 8.8.5. X(A,£) d

With the same assumptions as in 8.8.4, the set

is a closed, semi-algebraic set.

Proof: —————

If

A = R[X]/P,

P = (f.), then the simple points of I

are the points in the open, semi-algebraic set

X(A,$ ) w

{x|rank((8f./8X.) (x)) =codim P}.

(In fact, by looking at the possible subdeterminants, this is even an open semi-algebraic set.)

U{g.}, then take closure.

We intersect with

Tarski-

Seidenberg guarantees that the closure of a semi-algebraic set is semialgebraic.

•

Proof of 8.8.2. result„

Denote by

First, part of the proposition is a purely algebraic

Ax

the local ring obtained by localizing

and dividing by the ideal generated by the

fi#

R[X]

The assumption

at

x

df^Cx)

independent implies that the graded ring associated to the maximal ideal m

C A

is a polynomial ring, in particular, a domain.

a domain, or equivalently, (f1...f ) i r this says precisely that of

(f^.,^)

in

x

Therefore,

is a prime ideal in

R[X] . x

A

In turn,

is a zero of precisely one minimal prime

R[X], Geometrically, Z(P)

a Zariski open neighborhood of

x £ Rn .

and

Z(f io ..f r )

is

P

coincide in

For details of this argument, see

[63, Chapter 11], or texts on algebraic geometry. Now we must look more closely at small semi-algebraic neighborhoods of We must show that if

h

vanishes on

Throughout the argument we may assume assume of

P

df,(y)...df (y) and

Z{fi> H u ( g ) , U(g)

g(x) > 0, then

as small as desired.

h e P.

Thus, we

are independent, y £ U(g), and we assume the zeros

(^...f^) coincide in U(g) o

Suppose otherwise,

that is, P C l(Z{fi>

n

U(g)) = H p

where the

215

P.

x.

are convex prime ideals, strictly containing

P.

say

and by 8.8.1 we can find

x G Z(P ) . Then

y G Z(P Q ) n u ( g ) independent.

U

Z(P j ), such that

Since

we know that near Thus, f

codim(PQ) > codim(P)

G P

Z{£.} n u ( g ) = U (Z(P.) nu(g)) i j J y

the zeros of

df,(y),...,df (y). 1

f

o o

G

P

° *

=

(f,...f )

I

z

f

^ ^ i^

nu

f

y £ Z(P.), J

and

j t 0,

y, yet

df (y) 1

(Alternatively, if we choose by

f fJ

without changing

( g ) ) » which certainly vanishes on

€= n

f

yet still has differential independent of

p.,

j^o J

df Q (y).

Z{f i )

P .

is

° f

G P ,

coincide with the zeros of

Z{f.}, near

j,

are linearly

r

f Q (y) = 1» then we can replace f

x G Z(P-)> some

df Q (y) ,df: (y),. .. ,dfp(y)

vanishes identically on

independent of

We know

near

Now, y,

From this,

df,(y),...,df (y).)

we will derive our contradiction, by appeal to the implicit function theorem 8.7.2. By translation and linear change of coordinates, we may assume

and

f i = x i + (terms of degree •> 2) G R[X],

theorem states that near graph of a map y

i

= y

0

R n " r •* R r

i^ x r+l* ''Xn-^

are

the surface

f^ = ••• = f

The implicit function =0

• • -\h

where

algebraic functions of the last

n-r

variables.

Moreover, the tangent plane of the surface is the plane equivalently, (8yi/8x.) (0), function with have

df Q (0)

l < i < r ,

independent of

(8f /dx.)(0) £ 0, some o j

shows that

fQ

fQ

x, = ... = x

r+1 rn)

By the Nullstellensatz we can find c ij (y 1 ...y d ) + 0, and

i-ai(y1.. . y ^ t 1 " 1 t 0.

Thus

t(y 1...y d ) G R

G

Z(P) H U(g)

3f / 3t (yx . . . y d )

=

is a simple root of the

f (yx,... ,yd,T) 6 R [ T ] ,

polynomial Suppose

(z-^... zd )

will have a root f(z-...z,,T) id

x

is very near

near

T = t(y,...y ) . in

•

Tnen

f

(z^ • • • z d > T )

If

d+1 < j < n, define — — z = (z.....z ) £ R .

z G U(g) C R .

We are finished

z G Z ( P ) C R n.

the kernel of

f(xr..xd,T).

R

by the rational formula above and consider

Watching our estimates carefully, we may assume if we prove

in

(y^'-'y^)

tCy^.-y ) , by a simple estimate argument showing

changes sign near

z. = x. (z, .. .z ,,T)

Now,

Now, P

£ i GR[ X;L ...x d ]

f i i 0, and let

element for the field extension

over

contains a disc

'£w[g] -convex after all.

Conversely, if

field of

(equivalently,

W

TT(Z(P) flU(g)), which contains a disc, hence shows

CR[X]

'B[g]-convex primes.

1

tr.degR(R[x]/Pi) < d.

^[g]

contains a disc.

P = I(Z(P) n u ( g ) ) .

P CI(Z(P) nU(g)) = H P . , where the

x-^.-.x^ E A

e > 0 G R.

TT(Z(P) HU(g))

\[g]-convex if and only if

Let

be the projection

TT(Z(P) nij(g)) d

B(y,e) = {z G R | llz-yll _ 0, all (A(F),!P(F))

is x e F},

x e F } , then

is an RHJ-algebra (with, of course, X(A(F) ,'£(F)) = F.)

Strati-

fication considerations arise naturally in the proof of this result. E C Rn

Let

be any semi-algebraic set, and represent

E. = Z { f . . } n u { g I

IJ

Then the

I = U ^ P^

},f..,g GR[X ij iK l

and

n

X]. n

I(E) = 1(1") = n 1(1..)

are the minimal primes of

g i k £ P^ . hiQ e

iK

I(E^).

Let

I C R

n

E

as

E = U E.,

be the closure of

E.

Write I (Ei) = l(E/) = H p ^ , where Then

f. . E P. , all

j,a, and

This last holds since if, say, g i o E P^ , we could choose

Pi3-Pia-

Then

g i Q h i o e I(E i ), but

gio

is strictly positive on

pjFCX

E^, hence Let not have

h^o£

I(E i ), contradiction.

A. = R[X. ...X J1/P. , 'B. L 1 n / ia y i a ia

X(A. , tp. ) C E*., in general, because the XOc

10t

zeros far away from

1

ia' ^ia1 *

Proof:

Then

S

S.

= X(A. , 'JK ) , the non-degenerate points of

Moreover

> I^i^ J I^i " s i a ) •

We use 8.8.4, which characterizes

set of algebraic simple zeros

218

could have degenerate

However, we do have the following.

Let

iaC f i*

g.v IK.

U{g ik >.

Proposition 8.9.1. X(A

=r $ L 6[g.,1 C A. . Now, we definitely do 3 w ikJ la

x

of

P. , with lOt

S.

In

as the closure of the

g., (x) > 0. IK

Call this last

set

V. a , so

thus clearly Let

S. a = V. a.

V i a C E. = l{£^}

gi

n U{g ik >. Thus

= n gik.

Then

g^A^ x

Via = (

algebraic simple zeros

A

E

minimal primes of the inclusion

n u

to be

A

^i* ia^

with

vanishes on We have

and

S.a = V i a C E.. denote the set of

A. = 2 det(A ) 2 , where ia p p

x £ Z(P i )

We are now in Fat City.

{A }

Z(P. a) C Z{f..}

r x

where

g^OO

Via

> °>

as

Moreover,

is the set of above.

1". - V^ , hence on

I(E) = 1(1) =

r

r = codim(P. ) . la

/ O G A i a , that is, g i -A i a £ P i a .

ia^ia)

gj_#A^

it is obvious that

dvnens'ion of

Let

(3h./3X,), and let j K

it is clear that

gj_'A^ae15

Since

IL - S^ .

H p. . i,a i a

•

Define the

max dim(P. ) . This is the maximum dimension of the

I (E), which all must occur among the

A(E) = A (I) "• II A i a i,a

S = U s.ia C U E .I = L

with

f.. €= P.^ we have

P i a = (h.) C R [ x i . . . X n ] .

submatrices of Let

Since

and the order

Then,* by J 8 . 6 . 5 ,> each

X(A ia ,(!p ia ) d ) = S. a

and

P. .

A(E) H n($ i ) .

v(A.i a » ,('£• w10LJ)£>)



By 8.2.9,(A(E),A(E)

is an RHJ-algebra, with X(A(f),A(l) H (tp. ) ) = S.

Obviously,

nnflj.^)

A(E) nn(tp. ) =

icx ^

the functions in Consider

A(E)

nowhere negative on

E - S.

By Tarski-Seidenberg

Clearly, F - S = U I. - U S. i 1 i,a l a But

I(f.) C i(f. - U s . ) . I :£ i a ia

must properly contain some < dim(E.), hence

ia ^

S. E -S

is a semi-algebraic set.

C U ( Ix. - U Si.a ) , thus i a

dim(¥- S) < max dim (IT. -US. ) 1 ~ a ia

In fact, by 8.9.1, each minimal prime of P. . Thus, we have strict inequality

dim(E - S) < max dim(E.) = dim(E).

X(A/p) d C R n , where

dim(E. - U S . )

We can now repeat the

whole process above, beginning with the semi-algebraic set many steps we succeed in writing

1(1". - U S . ) I a la

E - S.

After finitely

E C

A = R[X1...X ]/P,

P

prime, and where

$ C A

is a

finite refinement of the weak order. As consequences, we have proved the following two results.

Proposition 8.9.2. (A(F),$(F))

F C Rn

be a closed, semi-algebraic set.

Then

is an RHJ-algebra.

Proposition 8.9.3. type.

Let

In order that

Let

•

(A,$) €= (PORNN), with

(A,«p) — (A(F),$(F))

A and R-algebra of finite

for some closed, semi-algebraic set

F, it is necessary and sufficient that there exist (1) finitely many primes

219

P. C A %

C A

with

i

(0) = n p .

= A p

/ -[»

and (2) finite refinements of the weak order

such that

$ = A H Il(^i) , under the inclusion

A -^IIA^ D

Proposition 8.9.3 should be compared with the results of 8.2, especially 8.2.11, 8.2.12, and the last paragraph of 8.2. '^

Working with derived orders

instead of ty , we can now deal with all closed, semi-algebraic sets, not

just the closed semi-algebraic sets. If we combine the Tarski-Seidenberg theorem, the going-up theorem for semi-integral extensions, and 8.9.2 we can prove the following.

Proposition 8.9.4. inducing set.

cp:

Then

R

n

m

-> R .

cp(S) C R

Proof:

m

Let

cp*:

Suppose

R[Y.....Ym] -•R[X1...X ]

S C R

The image

A(cp(S)) ->A(S), where

cp(S)

is a closed, bounded, semi-algebraic

is semi-algebraic by Tarski-Seidenberg and The homomorphism cp* induces an inclusion

A(S) = R [ X r . .Xn]/I(S)

Moreover, ^(cp(S)) = A(cp(S)) H ^ ( S ) .

Since

and

A(S)

is certainly semi-integral over A(cp(S)).

RHJ-algebra, hence by 8.2.7, so is

Remark.

A(cp(S)) = R ^ . . .Yj/IfcpCS)) .

is semi-integral over By 8.9.2, (A(S),$(S))

(A(cp(S)) ,$(cp(S))) , and

Thus, cp(S) C R m

X(A(cp(S)),'^(cp(S))).

be a homomorphism,

is a closed, bounded, semi-algebraic set.

bounded by simple estimate arguments.

A(S)

n

R, is an

cp(S) =

is closed.

D

As an immediate corollary of 8.9.4, we get that any polynomial

function on a closed, bounded, semi-algebraic set assumes maximum values.

This

generalizes the discussion at the beginning of 8.7, where we proved this result for bounded, closed semi-algebraic sets. In many ways the non-degenerate sets integral domain over

R,

$ = $ [g-] C A

X(A,-.p)d, where

A

is a finite

a finite refinement of the weak order,

w j are more natural "building blocks" for semi-algebraic sets than the Although

X(A,'p),

X(A,*£).

is not a manifold, as we have seen it is the closure of a

d-manifold, where

Specifically, it is the closure of the

set

x

V

d = tr.deg D (A). K of algebraic simple points

of

X(A,'B )

w implicit function theorem of 8.7 guarantees that that is, locally like

220

R .

Thus, X(A,$),

with

g• (x) > 0, and our

j V

is an algebraic d-manifold,

is a sort of closed "d-manifold with

boundary and singularities". contractions to fractions

If

K

E

of

A

Secondly, these orders

'£, C A

are exactly the

of finite refinements of the weak order on the field of

A.

Thus we can view

X(A,$,)

as an "affine model" for

is a semi-algebraic set, the procedure above for obtaining

E~ = UX(Ai,>|5i) , where

Pi C A ( E )

are primes,

A i = A(E)/P i , and

^

C A^^

a finite refinement of the weak order, definitely depends on a specific presentation of

E = UE-,

E. = Z{f..} n u { g . v } .

1

1

1J

It is also not really a

lK

stratification of E, since the pieces

X(A.,$.) can overlap, and, in fact, 1 -1- d

can overlap on more than their "boundaries", as in Figure (a). Even if they overlap of

only on boundaries, we may end up with a very unnatural decomposition

E, as in Figure (b).

it Figure (b)

We can rectify these problems somewhat by replacing the orders

(•£.) 1

by finite intersections of such orders.

Specifically, let

{P }

C A. d

1

index the

distinct primes which occur among the P.. We assume our decomposition E = U X(A.,'£.) is irredundant, in the sense that no term can be omitted. 1 -1- d On

A

a

= A(E)/P , we impose the order a

n p p

'£• = '£ . i a

Then we have

1

(•£ ) = fl(p.) 1 d d RHJ-algebras

and

X(A ,$a ) = u X(A.,$.) . i : L ^ d p = p d i a

(A , ('£ ) )

are geometric invariants of

= (A(E),A(E) if

a / 3.

A(E)

The primes

E,

E = U X(A ,$ ) ,

dim

) , and

The proof is not hard.

certainly minimal primes of primes of

Now, we claim that the

P

of maximal dimension are

A(E), hence are characterized as the minimal

of maximal dimension.

which is a geometric invariant.

Now

Moreover, X(A ,'£ ) = (E n X(A $ )) , i Y d "Y w d 1" - U X(A $ ) has strictly lower

221

dimension than invariant.

E.

Thus, by induction, the whole decomposition of

E

is

This discussion indicates that very nice basic building blocks

for semi-algebraic sets are affine models of orders on function fields where

'£

is a finite intersection of finite refinements of the weak order.

We can also use the ideals of degeneracy stratifications.

By definition, I

=

lim

= I(E n U )

x G E" C Rn.

if

U

Write

minimal primes of

C A(E),

I

x G E, to study

I(E fiU), where

smaller and smaller open neighborhoods of I

'£ C K,

x.

U

parametrizes

Of course, the limit stabilizes

is small enough, say a small ball centered at

I x = I (I H U ) = H P

where the

P i x C A(E)

are

I .

Proposition 8.9.5. (a) over

E". Each (b)

the

Only finitely many distinct prime ideals

£

a

If

C A

a

P. = I , ix y

for suitable

E = U X(A ,$ )

y

near

P.

P ot

The subset

is an irredundant representation of

P

= kernel (A(E) -*• A )

X(A ,•£ )

C E

is a minimal prime of

Proof:

which occur are exactly the prime, y G E . x G E

such

I . x

P. .

x

in

In fact, we can find

P.

is

y G I" O U

H z(P. )

y G U,

XX

such that

E H n

X

entirely of simple zeros of

P. .

such that

Then

I Pix,

y f. Z(P. ) , ~J X

= Z(P. ) O IL XX

X

The finiteness of the set of all

nl)-convex,

XX

0

j- P. , and an open set

I(U

If there will exist algebraic simple

XX 1X

I

is characterized as those

For example, since each

we know that arbitrarily near

P.

E", where

The proof consists of reviewing the various results of 8.6, 8.7,

8.8, and this section.

zeros of

varies

x.

prime ideals which occur as ideals of degeneracy,

that

x

are finite intersections of finite refinements of the weak

order, then the primes

(c)

occur, as

consists

X

= P. . x G E, follows from (b) and (c).

These two statements can be proved readily with all the machinery at hand. For example, suppose Consider the

X(Ag,«£g)

E H U , and since

222

I

= P

is prime, and assume

which intersect

U H X(Ag,$g)

E H U.

P = I(E H U ) , y G U. Since

is Zariski dense in

P

vanishes on

X(Ag,$ w ), we have

P C p

But also, some

X(A,# ) i

of

P

in

will contain a whole neighborhood of simple zeros

T d

E H U, so

P

C p , hence

P

= P.

We leave the rest of the details

of (b) and (c) to the reader.

•

The ideals of degeneracy can be used to define local notions of rank and dimension.

Namely, if

I

= H p. , define

d i m ( E ) = max dim(P. ) A

J.A

rankx(E) = rank x (I x ) .

(Rankx (I) = rank{dh.(x)}, where J fications of

E

I = (h.)> x G Z(I).) J

based on rank and dimension.

{x £ E|dimx(E) >^ k} an open subset of

is a closed subset of

E.

E

We can define strati-

Note that for any integer

k,

and

is

{x £ E|rankx(E) _> ^}

We do not quite want to begin a stratification with

all points of maximal rank, since this set will have (possibly rather singular) boundary points due to inequalities defining

E.

to be the "interior points of maximal rank".

These are obtained as follows.

Take the irreducible components take only those points

y

X(A ,'£ )

Cf

However, we can define

of least dimension.

E.

We will have

but will terminate.

Then

which are simple points, lying on a unique

X(A ,'£ ) , and for which an entire neighborhood of to

E

I(E-E^ -*) 3 I(E)»

so the

y

in

Z(P )

belongs

process can be iterated,

Using the implicit function theorem of 8.7,

E^ *

is

a manifold.

In the figure above, E^ ^

consists of the circle minus the vertex of

the triangle.

Then at the next stage

the triangle.

Next, E ^

triangle and finally

E

E ^

we get the interior points of

consists of the interiors of the edges of the is the set of vertices.

223

(x 2 +y2 )(y2-x-l) = 0

y 2 -x 3 + x 2

Figure (a)

Figure (b)

We point out that by defining rank locally, our stratification by rank is not the same as that of Whitney [44] for algebraic varieties

Z(I) C R n.

In our stratification, the origin has rank 2 in both varieties above, hence is the first stratum. origin has rank

0

On the other hand, in Whitney's stratification, the

in Figure (a) and the curve has rank 1, whereas in

Figure (b), the origin has rank 2 and the curve rank 1. 2 the polynomials

x(y

(In Figure (b),

2 - x - 1)

and

y(y - x - 1)

vanish on the variety,

but have independent differentials at the origin.) We can also find a manifold stratification of

E

by dimension,

of E E D E. Q . 3 E.j. D ••• . We begin with the pieces X(A ,'£ ) of greatest dimension, and take for the "interior" simple points, lying on a unique

X(A ,$ ) .

Then

dim(E-E, 0 .)< dim(E), and the

stratification continues inductively.

8.10. Krull Dimension Let

A

be an integral domain, •]} C A

an order.

By a weak %-oha-in of

prime ideals we mean a strictly increasing sequence of '^-convex primes (0) C P 1 C ... C p^ C A.

The length of the chain is

r.

We define the

weak Krull d-imens'ion, dim (A,$) , to be the maximum length of such a chain. In complete generality, this notion is probably uninteresting. we make the drastically simplifying assumption that

224

A

Even if

is finitely generated

over a ground field example, if then

R[T]

R[T]

R, the order

•£

is ordered with

can make things complicated.

T

For

infinitely large relative to

is a semi-field, with weak dimension zero.

R,

This pathology is

caused by lack of finiteness conditions on the order. Suppose, then, that we begin with a finite real domain R

real closed, and an order

the weak order. Thus

If

Q Cp

*jp = *J3 [g.] w 3

primes of length

any '^-convex prime, we can apply 8.4.3 to '^-convex primes above

Q,

Q C p

=£

l

R,

tr.degR(A/P) < tr.deg R (A/Q).

But from 8.4.3 we have chains of ^-convex

dim (A,'£) = tr.deg(A).

r, thus

over

obtained by finitely extending

are prime ideals, then

dim (A,'£) (A/Pr, tp/Pr).

From Proposition 7.7.9, a strong

#-chain requires much more restrictive convexity hypotheses on the primes

P^,

and by 7.7.10, a strong «j)-chain corresponds to a sequence of signed places K -• Kj^, ± «>-• ... "* K r , ± °° over where

K

dim (A,$)

is the fraction field of

R, finite on A.

P i = kernel (A -• K i ) ,

We define the strong Krull dimension

to be the maximum length of a strong $-chain.

actually produces strong '^-chains in tr.deg(A).

A, with

Of course, 8.4.3

A, hence we still have

dims(A,'.p) =

But what is more important, we have available the signed place

perturbation theorem 8.4.9 and its consequence Proposition 8.6.10 which gives more delicate information immediately.

225

Proposition 8.10.1.

Let

Q C A

there exists a strong 'p-chain

be a

(0) C P

'^,-convex prime

C • • • C P

= Q C • • • C P

'=f=-

' =F

=p

•$ = $ [g-]. Then

'=fz

^

C A, where

^

s = tr.deg(A) - tr.deg(A/Q) = codim(Q).

Proof:

The chain below

Q

K -» K-j^, ± oo-> ... -^ K g , ± «>.

comes from 8.6.10, which gives signed places

Then

Kg

is still a function field, to which

we apply 8.4.3, and extend the strong $-chain above Actually, a little more is required if some that the total order one finally obtains on one constructs an

integral extension

A^B

necessary in order that the weak order on Q

lifts to a convex prime of

field of

K

B

Q.

g. G Q

in order to conclude

actually extends $.

Specifically,

by adjoining as many

fgT

contracts to

The prime

$ C A.

as

B, and we apply 8.6.10 and 8.4.3 to the fraction

B, then restrict to

K.

This argument was actually used in the proof

of 8.6.10.

•

The reason strong dimension is a more natural concept than weak dimension is that strong dimension is a looal concept, giving geometric information about a semi-algebraic set infinitesimally near a point. prime x

of 8.10.1 is a point

Q

is a non-degenerate point

then any !pd-convex prime of

x

in

P.

in

x E X(A,$),. is

Then

P.

'^,-convexity of

Moreover, if

p[g]-convex.

X(A,'p)j, the prime ideals

have lots of zeros. of

P

x E X(A/p).

Specifically, suppose the

g(x) > 0,

says gGA,

Thus in any neighborhood

U

of the strong *p-chain of 8.10.1

Specifically, P. = I(Z(P.) n U ) , we can find simple zeros

U, and so on.

The strong chain of primes going down from

correspond to a chain of subvarieties going up from the point l,2,...,r.

Q

Q

x, of dimension

In other words, i/nfinitesimally near* x, we can move about on the

semi-algebraic set

X(A,'£)

with

r-degrees of freedom.

By way of contrast, we reconsider weak dimension.

The result which

allows the "desired" conclusion is the following.

Unproved Proposition 8.10.2. Then there exists a fpj-convex prime tr.deg(A/Q) = 1.

226

Suppose

Q C A

Q 1 , with

is '^-convex, $ = •£ [g.].

Q1 C Q

and

tr.deg(A/Q') -

Corollary 8.10.3. in A.

Suppose

Pi

C ... C p . _ C A

(0) C p^ C •. • C P^ C A

Then there exists a ^-convex refinement

of

r = tr.deg(A).

maximal length

Proof:

is any $ -convex chain

It suffices to insert an appropriate chain between any two

•^-convex primes we may assume

P C Q.

Passing to

(A,$) , where

A = A/P,

= tp/P = $ w [g.]» j Q1 C Q

P = (0). Apply 8.10.2 to find a $,-convex prime

dimension one greater than that of

Q.

of

Then apply 8.6.10 to go down from

Q'.

•

A proof of Proposition 8.10.2 can be extracted from [21]. Here is a rough outline of the geometry involved. R

By extending the ground field from

to an appropriate transcendental extension, we may assume that

dimensional, that is, a point. of points.

Now, we know the variety

X(A,$)

For example, if tr.deg(A) = r, near a simple point

Q

is zero

has lots X = X(A,'p)

looks like affine r-space smoothly embedded in some higher dimensional affine space point

Q.

Rn.

We consider sections of

X

by

n-r+1-planes through the

The implicit function theorem guarantees that many of these sections

will be 1-dimensional semi-algebraic sets.

(Simply take the

general position with respect to the tangent r-plane of The difficulty is, the point Y, and

X

n-r+1

plane in

at a simple point.)

Q may be an isolated point of such a section

Y may not be algebraically irreducible.

However, the method of [21]*

is essentially to argue that generically these sections irreducible 1-dimensional sets through

Q, although

Y

are algebraically

Q may indeed be a

degenerate point. Such a set corresponds to the desired prime

Q1 C Q .

227

This

Qf

£,-convex since, by construction, Qf has sufficiently many zeros in

is

V Note the proof of this dimension theorem 8.10.2 uses completely different concepts than those required for the study of strong dimension.

This is because

8.10.2 really is not a local geometric result at all, but a global property of semi-algebraic sets. not be described by

Near a degenerate point

Q, the variety

X(A,$)

will

r-independent parameters; although globally we can pass

a curve, then a surface containing the curve, and so on, up to the r-fold itself, through the point

Q.

Also, we point out that the commutative algebra

analogue of the dimension theorem 8.10.2 can be proved using integral extensions and various going-up and going-down theorems for prime ideals.

In our real

setting this method seems to break down because of the special hypotheses needed in 6.4, especially in Proposition 6.4.2(b). The analog of 8.10.3 for strong chains does not follow routinely from 8.10.1. A

and

The reason is that given Q/P

a

P CQ CA

C£j/P) -convex prime of

strong ^-chains between

P

and

Q

with

P

a ^.-convex prime of

A/P, we cannot immediately construct

by passing to

(A/P, ($j/P) ) . The

finiteness condition on the order is lost by this process.

What is needed

instead is a more vigorous version of the signed place perturbation theorem 8.4.9, which was the basis of 8.10.1.

Proposition 8.10.4. over

R, and

:p' C A

small relative to

ideals of where that on

A.

if

induced by

r

Suppose

(0) = Pi

and

and all

K = R(x-,-..x ) P^.

r

x.

are infinitesimally are

P r i m e $'-convex

signed place

p:

is the fraction field of

are'^-convex, where

$ CA

K -*R, ± °° ,

A, such

is the total order

p.

The proof is by induction on

is quite small.

Let

we may also assume that the

r, there being nothing to prove

r. = tr.deg(A/P^.), so that

As in the proof of 8.4.9, we may assume

228

is a finite integral domain

C P " S '** S p # o ^ 1 i =£ =£ x s

Then there exists a discrete rank

p(x.) = 0

Proof:

A = R[x,...x ]

is a total order such that all

R.

r = tr.deg(A)

A

Suppose

A

r=r

is integral over

> r, > ••• > r .

R[x,...x ] , and

{x, = xv(mod Pi.), 1 < k < r.}, give a transcendence

base for If

A/P^., all r

o

- r, = 1, then the place l

the total order rank

1

A

£'

on

K

A

1

C K

= R[x.,y.], with

and the subfield

x.

and

y.

P^ • of

Af

fraction field of a quotient of

the convexity of the

Write

R[x,...x

i j K i r i , K

F

CF

relative to

R, the

A

A'

to

q:

+1]-..[xn],

Also, K1 is the

A'/P- )> hence by induction, K ! -* R, ± °° , preserving

and for

generate the prime

f. v (x v ) 3 > K- "•

is integral over

x, K

over

Eb ik (O...O)T 1

x, K

over

P- Ac j j

in

AQ . b j

R[x,...x ] . l r

Let

f. ,, with J>^

•£' C K.

Let

r

denote

F. C r be the real closure of R(x,...x ) , J i rj C ••• ^F-i- Since all x. are infinitesimally small x.

are certainly finite relative to

R(x1...x

) . In j p! = p f x .. (This Kv. i. . .x r j 3 i

A

of the place

p!(x i )

R(x 1 ...x p ) .

r

Thus elements of It follows that the

R(x 1 ...x r .), in the appropriate residue field,

with some

xa. . G F.. f. ,

is a root of the non-trivial polynomials

for

then gives

K, and let

are all algebraic over

k > r

K

r - r., > 2. I — o k > r., let

be a monic polynomial equation for

to our integral dependence relation for

if for

of

is the multiplicative set of non-zero elements

is the center in

and we can identify

Pj

Thus we may assume

are infinitesimally small relative to

p'-tx^)

R, we

$'-convex primes

follows easily from Proposition 7.7.4 and results in 7.6.) P^.

tr.deg(K') =r-l.

K, we can write

by 7.7.8.

1

fact, P^. C A 3

with

in

R[x1...x ] . We have our total order

a real closure of

is a discrete

Composition and reordering

{f. ,} J K ' k>r-

0 = Ea. ., (x ...x )f. ,

so that

j > 2.

S. C A

k > r = r , then o

coefficients in

A

R

represent the minimal polynomial for

] , then j

1

If

If

) C K

over

signed place

A/Pi. = R [ x r - . x r ][x r

R[x1...x, .] .

K1

(in fact, of

K -• K ! , ± °° -*• R, ± °° .

p:

f. , = f. (xv) £ P| x j,K i,k K j

of

r-1

P! ,

K -* A, ± °° defined by

infinitesimally small relative to

A', and they necessarily form a chain in

we can find a discrete rank

if necessary.

R(x,...x

is the integral closure of

can lift all the '£' -convex primes

the desired

x,...x

p', .: R(,x1...xriJ

place, with image a function field

Moreover, if 1

j , by simply rearranging

in

Applying the place K,

p!

k > r, we get that

Eai.-^(C1.. • C r )T 1

over

F..

Also,

)x11

we let

0

0 = Eb.v (x, .. .x be an integral dependence relation IK i r K R[x.,...x ] , then 0 is also a root of the non-trivial polynomials 1 r

over

R.

229

Let R

0 < e £ R

and let

over

0 < e. € F. 3 3

F..

to a field

where

L

by adjoining the elements

1 k j,k I j,i 2 2 to F., hence also relative to R, and since e. < x. , all these generators

T

of

so

L

B

is real.)

We replace

A

by the ring

are infinitesimally small relative to

The places

p!

are still defined on

L

R. since

R^.^x

J

center of

p!

on

B

is a prime

all our hypotheses hold for

B.

restricts to a discrete rank

r

K = L.

) C L.

l

The advantage in this:

Q^.

which contracts to

Since any discrete rank place of

The

3 P^. C A. r

place

Thus p:

K, we may as well assume

if we now construct any total order

L -• R, ± °°

A = B, '£ C K

such that the {x-...x } are infinitesimally small relative to R and the {f. , |r. < k < r} are infinitesimally small relative to R(xn...x ) , then J »^ J 3 necessarily all

x. 1

are infinitesimally small relative to

are infinitesimally small relative to generate

Pi.'Ag,, where

we have degree

E C r F, C E. 1

over

recall, f

S. = R[x,...x r ] . Thus if

be the real closure of

E, which can be written

R[x x .. • x r _ 1 ] ,

(x ) £ P^

x .

pn f

.

If

D

E

so that

D £ (x )

is the place

E(x ,y) r

or

Since ...x ) E(f

of transcendence

, z ) , where, i ,r

1

on

K.

p

D = DEOr)

x of

over E

is non-trivial, hence

We are then finished by induction, We propose to reorder

E(x ,y)

is algebraic and so that all our hypotheses on

K C E(x ,y) are preserved in the reordering. It is better to work with the transcendental element

230

r - r 1 >_ 2,

represents a minimal polynomial for

as in the second paragraph of this proof. over

E(x ,x

is algebraic, the place

is discrete rank

f. , K 3 >

$ , we still have that the

x i = xi(mod P i ) . We consider the cut

defined by

p.

R(x 1 ...x r p .

Consider the function field

= £-

and all

R C x , . . ^ ^ ) . As we observed, {f. ,} 1 X 3 J » k k >r-

pDr . on K relative to this (new) order K^X]^. . .X r ) center of p. on A is Pj_.c A. J J Let

R

(x r) G E(f 1 ^,z)

= E(x , y ) , since

f,

R(x 1 ...x r ) C E.

Thus let

sign of

f,

is infinitesimally small relative to the (large) field D 1 = D £ (F 1

if necessary.

so that if

E(fn

)

E(f1

R(x,...x

We may assume

0 < D', by changing

We appeal to 8.3.1 to choose

is reordered over

new order extends to relative to

).

r

E

with

, z ) . Of course, 3

D c (f 1

3 £ E,

) = 3,

0 < 3 < D',

then this

is still infinitesimally small

).

We now consider our hypotheses on the elements

x.,

i H X d ^ 0, where

U{g.} n x,

R.

are Zariski dense in

(K) ,

'^[g^]

U{gi> = {x E x|gi(x) > 0}. X^.

We will find it convenient to forget the degenerate points of and work in

X^.

Thus, the notation

U{g^}

now means

Let

V

Vlg^} ^ ^

(In

Xd,

W{g i }-U{g i >

% [g^]

V{g^} =

has no interior.) V{g.}.

be the collection of non-empty sets of the form if and only if

altogether,

{x £ Xd|g.(x) > 0}.

We will also find it convenient to work with the larger open sets U{gi> = W{g i> C X d .

X

is an order on

A.

The

Vfg^}

Thus

are regular

o_

open sets, that is, V{g.} = V{g.}, since, in fact, V{g.} = U{g.}. Seidenberg, the

V{g.}

are open, semi-algebraic sets.

in any semi-algebraic set) the the

U{g.}.

That is, each

finite union of

Vs.

V{gi>

V

is a finite union of

U's

much the same statement as Unproved Proposition 8.1.2. fact, although there are strong indications that the

By a prefilter

V

£ &.

U

is a

In fact, it is pretty We will not use this V{g.}

provide a more

we mean a non-empty subset of ~//, closed under

A prefilter is a filter if

V G ^ ,

V C V ' G ^

implies

A filter is an ultrafilter if it is not properly contained in any

other filter in

X

Note that

This is the property of the U{g.}

and each

U{g.}.

^^- ^

finite intersections.

In affine space (hence

form a base for the same "topology" as

This is certainly not obvious.

natural base than the

By Tarski-

V{g i> C v ( h )

V{g i >

if and only if

hCVCg^) _> 0.

which makes them more convenient than the

for the purposes of this section.

The set of filters in

V

is partially ordered by inclusion, an arbitrary

intersection of filters is a filter (every filter contains a union of a chain of filters is a filter. an ultrafilter.

Each prefilter

^ C ^

X^ = V(l)), and

Thus every filter is contained in

is contained in a smallest filter

d Let if

$ C A

g. E •£.

only if

«^('p)

is a prefilter since

Define ^tp) C f tp [g.]

by

V{g.} e^"($)

is an order on

A

if and

V{g.} £ ^ , which implies the desired finite intersection property

for ^ p ) . If Let

be any partial order.

&CV

^

C $ 2 , then

an order on

2

be an arbitrary prefilter in

if there is V Cv(g).

p

V £&

Since the sets A.

with V

g(V) :> 0.

V.

Define

)-

V.

Otherwise, f

U{g.} ^ U ( - g ) , where

$ W

=

V = V{g.}.

'£(«£§)» where ^

If

is the filter

generated by &". We will study the compositions p ( ^ W ) and

&^&(${&)')

and ^^(0)).

First, $

are completely obvious from the definitions.

Proposition 8.11.1. (a)

If

«^C /' is any prefilter, then ^ ( ^ ) = ^

(b)

If £&($(&)),

let

is a filter, V(g.) e ^

V{g i ) G J T and (b) Suppose f = #(£}

admits a proper refinement, $ C'£[g].

and since ,F is an ultrafilter, 3~= g'&ig]).

Then But now

, contradiction. Secondly, assume

'^ C A

properly contained in a filter

is a total order and suppose (

S. Let

? C^C^QJ)) Ctp(®, we have fl = ?G?).

&- &($') is

V = V{g.} G ^ - ^ T . Since But

g± e ? ( ^ ) , hence

V{ g i >

contradiction. (c) Finally, ^ c'p(^(^)) and $(^f(^)) ).

Also, 'p, = fl !p

total order refinements of $. c

is a derived order, so

where the intersection is taken over the Write $ n

Then ^ p ) C ^ , fl(^CP)) ' P ( ^ )

cn

= $C^T)

where

= U{gi>

is

is at infinity. in

Xd.

If

^C

V

we mean

n v{g.} ex, x d .}e^ If

^ = ^ ' C ^ w [ g i ] ) , then

center of

&

C(0) = V{ g i >.

could be empty, even if &

In particular, C { M £ W ) ) = X d . is bounded.

The

In the case of filters

at infinity, homogeneous coordinates and hemispherical models of semi-algebraic sets makes these notions quite analogous to the centers of places in algebraic geometry over algebraically closed fields.

This is especially true for total

orders, because of the intimate relations between total orders and real places. In the real case, we have the added geometry provided by partial orders on function fields.

Proposition 8.11.2. &CV

of subsets of

order refinement £P =^*CP»), then

If $ C K

X d , then

'p! D $

with

C ^ " ) = x.

is an order with associated filter

x € C(^j x

if and only if there is a total

a 'p1 -convex maximal ideal of

In general, if

&

A.

If

is an ultrafilter, C(^)

consists of one point or is empty, and is always empty if

&

is at infinity.

235

Proof:

The main point here is that if

g £ A,

necessarily positive in any total order on ideal.

K

for which

$ [-g] C A

in

X H W(-g)

of

X H W(-g).

if

is an order and

g(x) > 0

x G V(g), then

and even if

g(x) = 0 ,

The proposition is then proved as follows. family of elements

g £ G A, all

e > 0 G R, with

This is easy using the technique of 8.1.1. #[§£> all

e

]

is

an

exists with

x

'£[g ] .

f

£ -convex.

Otherwise, x GV(-g.)> some

Then

j , and

g.

If

x

g

is

is a convex

x

For

is clearly not

is a degenerate point

X E Cf^O, we choose a

g (x) > 0,

The assumption

order, clearly with center

any total order refinement of

•£' 3'£

x

This assertion can be deduced from either 8.6.5 or 8.11.1.

example, if

'p'

x E V(g), then

{x}.

n v ( g ) = {x}. x G Cffi

implies

We can then take for

Conversely, if a total refinement x G V { g . } = O V*(gi)

all

would be negative in

$'.

g-,,...,g^

The other statements of the proposition are also easily deduced from the remark in the first paragraph of the proof.

Remark.

•

If our ground field is R, the real numbers, then a compactness

argument implies that the center of a bounded filter is never empty.

Remark.

If the order

$ C A

is finitely constructed using the operations

of finite extension, finite intersection, and the operators of 3.12, then

C(^('^)) C X,

will be semi-algebraic.

£ , !p , # , $ ,

In general, however,

this will not be expected.

Remark. • *

If $ C K = R C x , . . ^ ) in

the associated signed place, with center

{(a,...a )}

A

if and only if

is a total order and Archimedean over

pD: K

K -» A, ± °°

R, then &ty)

has

P R (^ i ) = a^ G R C A.

It is pretty easy to see intuitively (but not necessarily easy to prove) what all orders are like on function fields in one variable.

236

An affine model,

with degenerate points excised, will look like a smooth curve, with finitely many singular points. A branch at

x

At each point some even number of branches comes in.

is a connected component (see 8.13) of

a very small open neighborhood of

x.

U - {x}

where

U

is

The total orders correspond precisely

to the branches at all affine points, together with a finite number of branches at infinity.

The individual sets

V{gi>

are finite unions of

open intervals on the curve, with disjoint closures. In 7.7 we studied places if

A

is real and

p:

K •> A, °° of real fields

K.

In particular,

A

is given any total order, then by 7.7.2

p

can be refined

to a signed place, p:

K •*• A, ± °° , thus inducing an order on

K

compatible

with the order on

To conclude this section, we study a finite variant

A.

of this problem of lifting orders in the case of function fields. Suppose A = A/P, and '3 C A P

A

is a real finite integral domain, P C A

$ CA

such that

will be

a total order.

TTCP) = $"> where

a prime ideal,

We ask when there exists a total order IT: A -• A

'^-convex, hence necessarily

P

is the projection. is

($ ) -convex. w d

In particular,

This condition

will be included in our proposition below. Let

X = X(A,'£w),

X = X(AyB ) , X . C X W

X^ C X

the non-degenerate points, and similarly

the non-degenerate points.

We first prove a lemma

Q

of independent interest.

237

Proposition 8.11.3. X = X(A,£ w ), in

X

E C X

Suppose

A

is a real finite integral domain

a semi-algebraic subset.

Then

(that is, I (E) = (0) C A) if and only if

E

is Zariski dense

E H ^

has non-empty

interior.

Proof: dense. some

The "if" statement is clear since open sets in

Write E^^

E = U E^

Ei = Zff^} n u { g i R} .

is Zariski dense.

Thus, f i . = 0 e A

I(X-X,) j- (0), we must have

Proposition 8.11.4.

^{g.^} n X,

If

P C A

then there exists a total order all finite sets in

{f } C $,

and

is Zariski dense,

E_L = U(g i k ) H X.

Since

non-empty as claimed.

is prime, f C A

$ C A

E

are Zariski

with

f. £ 0, the set

= A/P

IT (•£) = $

•

a total order,

if and only if for

U{f\} H x"d n x d

has interior

X,. d

Proof:

First a comment on notation.

Also, we write

U{f;[} C x,

F = {f G A | ? ^'F, ? 7* 0}. is

Then if

X,

U{? i> C x , if Suppose

$

We have

X CX

f± e A,

Tr(fi) = 7 ^

= $ W [F]

Obviously, fr(t|J1) = $ C A . *

CP-i) -convex. A d

TT(^) = '^5, since

is already a total order.

order and

f C A P

is

exists, certainly

IT: A ->A. Now, let

is an order and

P CA

From 7.7.3, we can choose a

£ ^ ^

total order

P

A

total order refinement '|5 C A

such that

c

from

is 'J3-convex.

Then also

Conversely, if our desired

F O p , hence

•£, = '£W[F] C A

is an

(#..) -convex.

We now prove the hypotheses of the proposition are sufficient. prove

$.. C A

is an order and

order, then some

^ [£^]

P

is

If

'J3, is not an

is not an order, for finitely many

is equivalent to

U{fi> H X d = 0, so

U{f-} n X,.) i (X

'p, l

If

($,) -convex.

is an order, but

U{? i ) n ^ P

n X d = 0.

is not

We must

f^ C- F. (Note

This

u^}

H x"d

('p..) -convex, then again Id

there is a finite set of (A, e p j f i l ) ^

f. G F with P not ('£ [f-]) -convex. By 8.6.5, 1 W 1 (J is an RHJ-algebra, with X(A, ( ^ [ ^ 1 ) ^ = U{f i> H ^ . (Here,

the bar denotes closure.)

But if

U{fi> H X d n f ^

is exactly the ideal of functions which vanishes on C U{f i } H X d H X = U{fi> n X d n Z(P). convex.

238

has interior in U{f.} n X, H X,

But in this case, P

would be

X*d, then

P

Finally, we prove the hypotheses are necessary.

Assuming

P

is

(•£ {f.}) W

convex, we know that

P = I(U{fi> H X d O Z(P)).

Zariski dense in

Since

X.

f = IT£.

That is, U{f i ) H X

vanishes on

f £ P, we must have

8.11.3 to

l

U{f i ) H x ^ f l x

A, we conclude that

Zariski dense in

U{fi> H x d H x"

1

a

X.

i

d

Applying

has interior in

x"d, as

desired.

•

Corollary 8.11.5. if

X

In the situation above, if

X = X(A,'J>w) = X d , that is,

has no degenerate points, then any total order

total order on

Proof:

'£ on

A

lifts to a

A.

The point is if

X = X,, then a

I

Gf,

I

f 0, then

U{?i> n X, + 0.

If

U{?.} n x . O X = U{f\} n l , I a d ! d

For example, if

D

A = R[X,...X ] , the polynomial ring, then 8.11.5 applies.

However, the conclusion admits a trivial proof in this case. the fraction field of

A = A/P.

give the polynomial ring small relative to C A[X,...X ] . of any

A

is

U{f.} H X, - U{f.} H X,

l

and

H X

A.

Then

A[X.....X ]

tp

A

induces a total order on

any total order with

denote A

(X. - C O

We then restrict this order to the subring

We see this order lifts

f G R[X][...Xn]

Let

and we infinitesimally

R[X,...X ]

'£, by writing out the Taylor series

in powers of the

(X± - q ) . Here

q

= X^mod P) G A.

The geometry of 8.11.4 is roughly illustrated by the picture below, where

we have orders

X = X,

for simplicity.

*& C A, we can think of

subsets of

X,.

Using the ultrafilter interpretation of total 'jf as

picking out infinitesimal ly small open

If these small sets do not contain enough points in

there can be no ultrafilter of sets in

Xd

giving an order on

A

Xd,

lifting $".

239

In the figure, orders

*p

"centered" in the right half of

X,

will lift,

while orders centered in the left half of

X will not. d We will state a slight generalization of 8.11.4.

Proposition 8.11.6. Suppose $

[g-1 *- A 3

w

c

A.

is a real finite integral domain,

a finite refinement of the weak order and

'•Pw[g-1-convex ideal. $ w [g-] j

A

Suppose

£ C A = A/P

Then there exists a total order

= t, i f and only if for all finite sets n

f^g.} nx(A,!D w[g.]) d X(A,-B w [g j ]) d

Proof:

P C A

a prime

is a total order refining !p C A

with

{?.} C $",

and

g.l

f\ ^ 0, the set

has interior in

First, 8.11.3 generalizes routinely to sets

Then the proof of 8.11.6 is exactly like that of 8.11.4.

X = X(A,£ [g.]). w 3 The reason one

must work with the closure of may belong to

P, that is,

U{f.,g.} H X(A;J [g-]) is that some g. 1 w J 3 d 3 g. = 0. If, in fact, all g . f- P, then one

can just require that

U{f.,g.} H X ( A ^ [g.]) meets X(A,'£ [g.]) in 1 W W 3 3 d 3 d a s e t with non-empty i n t e r i o r . For e x a m p l e , i f g . f- P and X(A,$ [ g - ] ) = w 3 3 d X(A,$ w [g.]), this always holds, giving a generalization of 8.11.5. D

8.12.

Discussion of Total Orders on Suppose given a total order

$

R(x,y) on

R(x,y), the rational function field

in two variables over the real closed field

R.

Let us assume that

'£

is

2 centered at the origin in pD: R

where

R(x,y)

R , that is to say, P

R

M

= P R (y) = 0 G R C A,

R(x,y) -* A, ± °° is the signed place associated to our order on

and the subfield

R CR(x,y).

The results of the preceding section show that the order describable by infinitesimal behavior of functions On the other hand, the order pR:

R(x,y) -* A, ± °° .

f £ R[x,y]

is necessarily near the origin.

is also described by the signed place

In this section we will reconcile these two descriptions

by comparing invariants of

240

$

*£

pR

(the residue field

A, value group

r, rank

of

Pn, etc.) with the particular geometric behavior of functions

origin which determines whether

f

is positive or negative (rel

f

near the

'£). We do

not attempt to state a theorem, and our discussion is meant to be enlightening, not rigorous.

Our discussion of signed places on

R(x,y)

is a watered down

version of the discussion of valuations on function fields of two variables given by Zariski in his work on resolution of singularities of surfaces. There are two possibilities for the residue field p •

R(x,y) -• A. ± oo #

Namely, A

Archimedean ordered over A ^ R. over

R, and

cut of in

There is then

R

R

A

R, or

converging to

T.

may be very large.

converge to have

is isomorphic to

0.)

R(t). Let

and let

(a!,aV)

This means

is either smaller than some {i}

A

R.

t = f(x,y)/g(x,y) G R(x,y), with

t

of the signed place

is either a function field in one variable,

is algebraic over

defined by

A

Assume first that R(t)

T = D R (t)

Archimedean

be the transcendental

be a nested family of intervals

a! < x < aV

and every element of

a!^ or larger than some

aV.

R

(The index set

Also,it does not follow that the differences

aV - a|

$ C R(x,y), we

In any event, with respect to the order

a! < t < a'.'. Consider the neighborhood in

R

defined by

U i j £ = {(x,y)|0 < x 2 + y 2 < e , a!g2(x,y) < f(x,y)g(x,y) < aV g 2 (x,y)}.

We decompose these

U.

into their "connected components".

are non-empty and have the origin in their closure. and

g(x,y)

vanish at the origin.)

These components

(In particular, f(x,y)

Moreover, the behavior (number and

approximate location) of these components stablizes for sufficiently small and

a!,a'.'

close enough to

T.

The order

one of the component families of the in the next section, 8.13.) if

h(x,y)

U.

$ .

is then determined by selecting (Connected components are studied

Specifically, h(x,y) G R[x,y]

is positive in

$

assumes only positive values on the selected component, infinitesi-

mal ly close to the origin.

We interpret this selection of a component of

as selecting a branch at the origin of the curve larger field

e

R ( T ) . If

A = R(x,u),

u

f(x,y) = Tg(x,y)

algebraic over

U.

over the

R ( T ) , this choice is

just the choice of one side of a real root of the minimal polynomial for

241

u.

As a concrete example, let

R

be the real algebraic numbers

component of a

Given any polynomial h

h(x,y), we can find numbers

has no zeros in the intersection of the cone

disc

0 < x

2

+y

2

< e.

of this region, hence In case h(x,y)

ry < x < sy

U.

e > 0

so that

and the puctured

Then by requiring, say, 0 < y, we single out a component h(x,y)

will have constant sign in this component.

A ^ R, the value group

is the order to which

r

h(x,y)

branch of the transcendental curve first non-vanishing derivative of Note that if

r < TT < s,

t = x/y,

is always

Z.

The value in

Z

of

vanishes at the origin along our chosen

f(x,y) = Tg(x,y). h(x,y)

That is, we find the

on the branch at the origin.

R = ]R , the real numbers,this example cannot exist since

]R

admits no Archimedean extensions. We now turn to the R-valued signed places are four possibilities for the value group

Case I.

r

non-Archimedean.

(m,n) < (m',n')

Case II.

if

r

Case IH(a).

m < m

1

T

242

There

r = Z 1

m = m ,

x

Z, ordered lexicographically,

n < n1.

That is, F = Z.

Archimedean, non-discrete, and containing two incomWe can then assume

an irrational real number.

F

R(x,y) •> R,± °° .

r.

Archimedean and discrete.

mensurable elements.

Case HI(b).

or if

Then

pn: K

F

F = {n+mx|n,m £ z ) , where

is ordered as an additive subgroup of

Archimedean, non-discrete, but a subgroup of

T

is

IR .

Q, the

rational numbers. the F

p.

The prime integers fall into two classes, p i ? q., where

occur with arbitrarily high powers in denominators of elements of

and where the powers of

say by

b..

q.

occuring in such denominators are bounded,

Then

a 1

= p

•••

r Pi

3, 3_ q / '•' *j . 0 < a. < oo, o < 3. < b. } .

We now analyze orders on

R(x,y)

which yield signed places

p:

R(x,y)-*R,± °°

with these value groups.

Case I.

Here we have a rank 2 valuation.

defines an algebraic cut of the subfield there is an algebraic function f.(x) G R[x]

with

irreducible in

f(x,y)

R(x) C R(x,y).

y,

In other words,

f(x,y) = fQ (x) +f 1 (x)y+--« +f m (x)y m ,

infinitesimally small relative to

R(x), and

R[x,y].

The curve

f(x,y) = 0

discussion in 8.11 or 8.3.

V

One of the variables, say

/

thus has branches through the origin, as in our The order

$ C R(x,y)

picks out not only a real

\ \ \ \ \ \

root of in

R

2

f(x,y) over

R(x) (which is a branch of the curve

f(x,y) = 0

at the origin, as indicated by solid lines in the figures), but also

a side of this real root in the real closure of

R(x) (as indicated by the

normal arrows to our branches in the figures). With these choices made, the £-sign of a function determined as follows.

If

g(0,0) > 0, then

g

g(x,y) G R[x,y]

is positive.

If

is

g(0,0) = 0,

243

but

g(x,y)

does not vanish on our branch

x

g( >y))> then to the origin.

g

(that is, f(x,y)

has the same sign as its values on Finally, if

small normal curve to the values of

C

g

g

vanishes on

does not divide

C, infinitesimally close

C, then

g

will not vanish on a

C, near the origin, and the sign of

g

is the sign of

on such a normal curve, on our preferred side of

C.

(In

fact, this last "test" actually covers all cases.) The value group

V

is

determined by first writing is, h h

x

z

0 =

2 a..x y , i+j=l l j

h

f ) . The invariant

vanishes at the origin, along

r = Z,

The value

g = f h, where

is not divisible by

Case II.

Z-

A = R.

(m,n)

assigned to

g

does not vanish on

C

n

is (that

is then the order to which

C.

In this case, there is an "analytic curve"

a.. £ R, with non-trivial real "branches" at the origin, 1J

and the $-sign of a function

g

selected branch of the curve.

is the sign of the values of

g

on a

Now, what does this mean in light of the

fact that convergence may not even be sensible, and even if it is sensible, a. .x1y-) = h(x,y)

2

may not converge?

The answer is provided by looking

13

i+j=l

a t t h e honest curves

0 = h r ( x , y ) , where

hr(x,y) =

r I

i

The point is the infinitesimal behavior of branches of origin will stabilize.

precisely because

g

h(x,y)

h (x,y) = 0

at the

g(x,y)

on these branches, near the origin.

is positive or negative will be stable.

It is

is not a polynomial, that our branch for

0 = h

differs slightly from that for the branches chosen for all

r.

the (stable) order to which

g

branches of the curves

r -• °° .

•*

We can coherently select branches in the limit, as

r -> °° , and measure the sign of The decision of whether

i

a^.x y J , as

i+j=l

0 = h .

0 = h , and no The value in

g(x,y) ^ 0 r = Z

can vanish on

assigned to

g

is

vanishes at the origin on the selected (In other words, the degree of the first

non-zero term in some power series.) As a concrete example, consider

2 n!x . Given g(x,y), n=l we look at the behavior of g(x,y) infinitesimally close to the origin, along the honest curves y = 2 n!x . As r gets larger, we must look nearer and n=l

244

h(x,y) = y -

nearer the origin, but "convergence" of

Case III (a). one example here.

h(x,y)

is irrelevant.

Y - {m+nT|m,n £ Z, T irrational}, Intuitively, the '£-sign of

A = R.

g(x,y) £ R[x,y]

yielding this value group is computed by restricting dental curve", for example, r./s-

converging to

T

y = x .

We give only

g(x,y)

for an order to a "transcen-

What one really does is find rationals

and then restrict

g(x,y)

to the curves

y

= x

,

or more precisely, to coherently chosen branches of this family of curves.

Case m ( b ) .

F Cj},

A = R.

Again, we give only one simple example.

We might have a series representation (mi/m) < (m2/n2) < ••• ^ F.

y = x *

2

* +x

2

+ ..., where

This representation is purely formal.

finitely truncated formulas define

The

honest algebraic curves with "stable"

infinitesimal behavior near the origin.

We test

g(x,y)

by restricting to

suitably selected branches. It is clear that the "general" total order on

R(x,y) is complicated.

This is consistent with our set theoretical characterization in the preceding section in terms of ultrafilters of certain open subsets. algebraist should only tolerate the orders of type I valuations on fields of functions in

r

variables).

In general, the

(discrete, rank

r

These are more in

line with our philosophy of finite algebraic computability.

The other types

of orders are perhaps interesting to analysts. For example, consider a first order differential equation Q(x,y)dy = 0,

P , Q G R[x,y],

Q(0,0) t 0.

P(x,y)dx +

We ask what interpretation can

245

be made, for an arbitrary real closed field say through (0,0)?

R, of the "solution curve",

Although we do not expect an honest curve, we do have a

procedure for deciding if a polynomial

f(x,y)

is infinitesimally positive

or negative at the origin, along our "phantom" curve.

One approach is to

00

just take the formal power series solution

order, as in Case II above.

y =

2 a.x i=l X

and construct an

However, it is much better to just use the

differential equation itself to decide if a polynomial ought to be positive or negative along this curve, say in the positive x-direction, near the origin. Specifically, first look at (df/dt)(0), where

know

is the evaluation of

f

f(0,0).

y = y(t)

If

f(0,0) = 0, then we want to

is the "phantom" curve and

on the curve

t > 0.

f = f(t,y(t))

This doesn't make sense, of

course, but the result of applying the chain rule does make sense,

Sf $$. since the differential equation says computation of order.

If

(df/dt)(0)

dy/dt = - P(t,y)/Q(t,y).

is positive, then

algebraic solution

is positive in our total

(d 2 f/dt 2 )(0), and so on until we

(df/dt)(0) = 0, we compute

finally reach a decision.

f

If this

Of course, if the differential equation has an

f(x,y) = 0, we are in Case I above, rather than Case II.

This sort of interpretation of differential equations seems quite reasonable and worth further study.

Equations of higher order and behavior

near singular points are topics to be investigated.

Of course, one wants to

make sense not just of the germ of a solution curve at an initial point, but also the continuation of the curve. one wants to associate an order

c

$ x

values of

xQ

in some interval.

o

That is, given an initial point,

R[x,y], with

D R (x) = x , for all K o

This collection of orders will play the

role of the solution curve of the differential equation through the initial point.

246

8.13.

Brief Discussion of Structure Sheaves The material in this section is partly in the form of an outline of a

discussion to be worked out in detail elsewhere.

On the other hand, the

ideas are quite fundamental for our program of algebraizing topology. Let

S

be a closed, semi-algebraic set, identified with the maximal (A,'£) = (A(S) ,»B (S)).

convex ideal spectrum of an affine coordinate ring

Recall from 8.9.3 that this situation is intrinsically characterized as follows. A

is a reduced R-algebra of finite type.

primes

P. C A

$• = 'P [g-J I

w

with on

A -* ITAi.

(0) = Hp., and finite refinements of the weak order

A- = A/P.

ij

The set

There are finitely many '^-convex

such that

*J3 = A HII0&.) , under the inclusion •*• d

1

1

S

is the union of the sets

S i = X(Ai,('Bi) ) , each of

which is the closure of the set of algebraic simple points variety

X(A.,# )

with

g.-(x) > 0.

non-empty strong open set We will refer to such

x

of the real

In the semi-algebraic sets

U{h.} = {y £ S-|h.(y) > 0}

S., every

is Zariski dense.

as -irreducible components of

S. = X(A.('J$.) )

S.

In Chapter V, we constructed rather generally a structure sheaf relative to the Zariski topology on

Spec(A,$).

The global sections turned out to be

the ring

(A CM >., ']) en J

than

Because of the Nullstellensatz, this amounts to inverting all

1.

functions

f £ A

obtained from

with no zeros on

S.

A

by inverting all elements greater

Also because of the Nullstellensatz

we can simplify the discussion by restricting this sheaf to the maximal convex ideals

X(A,'£) = S.

Zariski open set elements of on

A

D(f) C S

x £ S.

f £ A, the ring of sections over the basic

is

(Acr,~ , '£C/-.pO> obtained by inverting all

with no zeros in

D(f). Thus elements of are

are functions nowhere

D(f). The stalks of this sheaf are the local rings

(Ax>'£x)>

Elements of

neighborhoods of

x

Ax

^cr^

those functions in

Acr4~ Ag .f .

D(f). The elements of

negative on

If

may be regarded as germs of functions on Zariski open

and such a germ belongs to

where negative on a Zariski open neighborhood of On the semi-algebraic set

S

£x

if and only if it is no-

x.

we also have the "strong topology", that

is to say, the collection of open, semi-algebraic subsets

U C S.

We would

like to study "sheaves" for this strong topology, but classical sheaf theory

247

is intimately tied to the infinite procedures of point set topology, so we should proceed with some caution.

Perhaps the most natural finiteness con-

dition to impose is that we seek sheaves for the Grothendieok topology on the set of open subsets {IK -* U}

are allowed.

U Cs

in which only finite covering families

When discussing open, semi-algebraic sets and

finite open coverings, it is obviously very convenient to assume Unproved Proposition 8.1.2. of any sets

U C S

This assures us that, essentially, finite open covers

just amount to writing

U{f.} C S.

U

as a finite union of basic open

On the other hand, all the propositions proved in this

section are independent of Unproved Proposition 8.1.2. We have in mind three sheaves of rings, in fact, associated to any semi-algebraic set

E C Rn.

We call these sheaves the rational structure

sheaf, the semi-algebraic structure sheaf, and the smooth structure sheaf. Each structure sheaf corresponds to a category of morphisms between semialgebraic sets,

although we do not study these morphisms here.

These

three categories can be interpreted, within real algebra, as delineating the three subjects, algebraic geometry, algebraic topology, and differential topology. (I)

Let

E C Rn

be a semi-algebraic set, A(E) = A(!) = R f X ^ . ,Xn]/I(E)

the affine coordinate ring of embedding of

E

E.

Throughout this section the particular

in affine space can be suppressed.

invariantly as some dense subset

We can think of

E

n

E = UE.,

E. = Z{f..} U { g . v } C X (A (!) ,'£ (!)) ,

1

of the maximal ideal space of an RHJ-algebra

1

1J

(A,'£)

of a certain type.

IK

None-

theless, it will sometimes be convenient to refer to the distance between points in embedding. form.

Rn,

||x - y||. Thus we do not go out of our way to avoid an affine

The reader can reformulate for himself all statements in invariant

For example, if we say a subset

S C E

then this can be reformulated by saying that all

h £ A

are bounded as functions on

is closed and bounded in S C X(A,'£)

R ,

is closed and that

S.

The rational structure sheaf is very similar to the structure sheaf for the Zariski topology.

We first define it, then prove a few propositions

which enable us to compute rings of sections, in some sense. is (relatively) open, we define

248

I(U) = {f

E

If

A = A(E)|f(U) = 0}.

U C E Then

A/I(U)

is a ring of functions on

U

and we define

A(U)

to be the

localization of this ring obtained by inverting all functions with no zeros on

U.

We define an order

negative on

U.

•£ (U) C A ( U ) , consisting of all functions nowhere

We obviously have a presheaf of partially ordered rings

for our Grothendieck topology and the rational structure sheaf of the sheaf associated

E

is

to this presheaf.

In order to make computations, let us first consider the case where E = S = X(A,*0), where

A

is a finite integral domain over

R, and

is a derived order of a finite refinement of the weak order. $ = 0#,T[g.]) w 1 d Any non-empty basic open set U = U{h.} C s is Zariski dense, hence A(U) = A.

Proposition 8.13.1. order

$' = (:p[h-]), J d

an RHJ-algebra.

C

A

Let

coincides with

Secondly,

D(h) Cx(A,$»), where

U = U{h.} C S , as above.

U CX(A,'|5)

h=nh..

Then, first, the

('£w [g-,h.]) , hence ! J d

Proof:

C

C&Jg^V )d

m

i

j

(CVgi] ) d [ h j ]

w

j

I

since

h. > 0, J ~

II h. ^ 0 J

means all

}

= d

m _> 0}.

*[hj]V

BUt

is nowhere negative on

J d by Proposition 8.6.5.

tp [g.,h.]

It is completely obvious that

D(h) CX(A,'£ ! ),

£ f, rel •£' , some

also, i t is easy to argue that any f ^ 0B[h.]) U(g.,h.}, hence belongs to

which

Finally, (A(U),fl(U)) = (Ag ( h ) ^ s f h ) ^

S(h) = {f _ e}, part of

In this case, d g

is easily computed

explicitly:

fz(x!-x,) 2 if (x'...xM ,t (X....XJ if

256

(X-...X-) = (X....X1 .

In particular, it is clear that the graph of and, as already observed, has the

e-6

d R , above

property.

Thus

continuous by the first part of the present proof. a minimum value on establishes the

is semi-algebraic,

2 dfi:

F Q •> R

is

It must therefore assume

F , which it obviously does not do.

e-6

F ,

This contradiction

property for our original continuous function

f.

•

We have some immediate corollaries of 8.13.8.

Corollary 8.13.9. and

U C R

If

f:

E -* R

is a continuous semi-algebraic function

is an open, semi-algebraic set, then

open, semi-algebraic in

E.

Corollary 8.13.10.

If

z €= E

is (relatively) D

E

is a connected semi-algebraic set, f:

a continuous semi-algebraic function, then there exists

f" (U) C E

with

x,y G E, and

E -*> R

f(x) < t < f(y) *= R,

f(z) = t.

•

The following result will be useful in the last part of this section.

Proposition 8.13.11.

Suppose

semi-algebraic subset such that semi-algebraic function.

E

E Cf ,

fQ

Suppose

fQ:

E Q -* R

f:

and which has the

e-6

E "* R.

a

is a continuous

Then there is at most one extension of

continuous, semi-algebraic function which extends

is a semi-algebraic set, E Q C E

fQ

to a

Moreover, any function

property on

E

f

is, in fact, a semi-

algebraic function.

Proof:

The uniqueness of

continuous functions.

f

is obvious from the

Also, given an extension

property, it is clear that the graph of in

E x R

of the graph of

fQ,

f,

f

of

e-6 f

property of with the

F C E X R, is just the closure

F Q C E Q X R C E X R C E" X R.

We made use above of the distance function to a closed set say

dR:

R n "* R,

d R (y) =

e-6

min |y-b|. b £ B always continuous, semi-algebraic.

D

B C Rn,

We now want to prove that

dg

257

is

Proposition 8.13.12. :

R

n

-*• R

Proof:

If

B C Rn

is closed, semi-algebraic, then

is continuous, semi-algebraic.

The

e-6

property of

dn

is immediate from the triangle inequality

D

for distances.

Thus we only need to prove that the graph of

is semi-algebraic.

dg

Rn x R

in

To see this, begin with the subset

D = { ( x , b , ||x-b||)|x G R n, b G B} C Rn x B x R . D

is obviously semi-algebraic, hence by Tarski-Seidenberg, so is the image

of

D

TT: R n x B x R -• R n x R.

under projection

Now, ir(D) C R n x [0,co),

and the subset

D1 = { ( x , t ) | x G R n , 0 < t e R ,

is also semi-algebraic.

This again follows from Tarski-Seidenberg since

is defined by an elementary sentence. now be described as

[ 0 , t ] n ^(D) = 0}

1

(F

dg :

Note that the graph of

!

n

- D ) U (B x {0}) C R

x R.

Thus

dg

D'

R n -• R

can

is a semi-

algebraic function.

•

As a corollary of this argument and 8.13.8, we can prove that continuous semi-algebraic functions are uniformly continuous on closed, bounded sets.

Corollary 8.13.13. and

f:

S -* R

there is a

Proof:

If

S C Rn

is a closed, bounded semi-algebraic set,

a continuous semi-algebraic function, then for all

0 < 6 G R

such that if

For each

x,y G S

x G S, the set

S

—————

and

= S

X,£

0 < e £ R,

||x-y| < 6, then

= {y G si If(x)-f(y)I > e} is X

I

—

a closed, bounded semi-algebraic set, by Corollary 8.13.9 (possibly but the set of such

x G S

with

S

obviously any modulus of continuity points

x.

= 0 6

so is

258

Then

BQ

It is not hard to prove that

AS = {(x,x)|x G s} C s x S, say using the

= 0,

e-value at these

Anyway, the rest of this proof goes through even if some

B = B Q C s x s.

S

is open and can be discarded, since

will work for our

B Q = {(x,y)|x G S, y G Sx> C S x S.

Let

|f(x)-f(y)| < e.

e-6

S

=0.)

is semi-algebraic, hence S H B = 0, where

property of

f.

Thus, the

distance function 6'

on

AS.

dR:

AS -• R

assumes a minimum, strictly positive value

We are here computing distances in

no greater than the distance from if

0 < 6 < 6', then

for our given

e.

x

to

S

R n x Rn.

in

Obviously, 6 !

R , for any

then we

is any union of finitely many connected sets

can consider maximal connected subsets of the form

are routinely shown to be both open and closed in components of

E.

sets, we can write

E^ U ... U E• . x x l k

These

E, hence are the connected

If an E exists which is not such a finite union of connected E = Uj U uj

are open and closed in

E.

where

l^ n uj = 0

Then, say, U-,

splits

producing an infinite strictly decreasing chain and closed subsets of

E.

The sets

and both

l^

and

U|

U-, = U 2 ^ Ui, and so on, U^ ^ U 2

V i = LL - U. + ,

c

U- ^ •••

of open

then give an infinite

collection of pairwise disjoint, open and closed, semi-algebraic subsets of

E.

We refer to this process as a splitting process. Clearly, we need only consider E of the form Z{fi> n U { g . } C R n , where 1 _< j _< k, say. Then E is the image of the real variety V = Z{f.,y.g. I 3 j - 1} C R n x R k = R n + k , under projection and we adjoin new variables since if

260

V. C E,

IT: R n + k -+ R n , where

Xi***^-

Tnus

it:

f^g. e

R^.-.xJ,

suffices to study varieties,

i > 1, are disjoint, open and closed subsets, then so are

IT

(V^)

c

assume

V.

V

If some

V

is not a finite union of connected sets, we may

is irreducible and of least dimension with this property.

that any O-dimensional semi-algebraic set is finite, so Let

VQ C v

singular set. where

E^V

be the algebraic simple points and

Then

is connected.

closed subsets.

Each

We split

E.V

V = l^ U UJ

is either in

continues with, say, Uj = U*2 U U£.

U,

Those

Now, the conclusion we want is that

VQ

dim(V) > 1.

EV = V - V Q

dim(EV) < dim(V), so we can write

Note, if

U|

in

into disjoint, open and

or

Ui.

Z^V C ^

This splitting process

are either in

V^

U1

less than

contains all

as above.

l£J, then

of the

splitting process.

U£.

V\ ,

i >_ 1, which

U1 C V Q

is one such set.

U*2.

E^V

If the

U^ = U 2 U U', then we

If not, we must start over and go back and split

V^ = U^.

and some in k

or

R .

UJ, but at least we have are in

U2

actually contains infinitely many

splitting process continues infinitely, starting with get all our

the

EV = Z^V U ... U E k V,

pairwise disjoint (relatively) open, semi-algebraic subsets are actually dosed

Note

Another case to consider is if some

Then we haven't even found

in whichever of

l^, U'

EiV

V 1 , but there are

begins the infinite

It follows, then, that sooner or later the splitting

process gives infinitely many disjoint, relatively open in

V\ C v , which are semi-algebraic, pairwise Rn.

V , and closed in

We are now in a position to use Whitney's proof [44] of the finiteness of the number of components. V

and a point

cuts the manifold

x

G V,

V..

and

xQ.

Let

g:

V -• R

transversal ly at x

be the function

x .

V^

connected sets. arranged that

In fact, we can assume that

g(x) = ||y-x| 2 . V

is a closed, semi-algebraic set in

value at, say, x^ G v^.

Then

x^ G V , hence

V

of lower dimension.

V

If

I(V) = (f±)

= {x1 G v|rank(df^(x),dg(x)) = r}. R n,

g| v

assumes a minimum

is not a finite union of

On the other hand, by our choice of x Q f- V , hence

||y- x || centered at

lies in the tangent plane to V..

codim(V) = r, we consider the subvariety

Since each

We bhoose a point

so that the sphere of radius

the normal vector to this sphere at at

dim(V-) = dim(V ) > 0, since, in fact,

is homogeneous, in the sense that it is a manifold.

y GR y

We know

y

and

is a proper subvariety of

x Q , we have V, necessarily

This contradiction proves our desired result.

261

Proposition 8.13.14.

Every semi-algebraic set is a finite union of pair-

wise disjoint, connected, open and closed, semi-algebraic subsets.

Remark: of

E

A further result is that if

which contains

x

is the set of

semi-algebraic "path" p:

•

x G E, then the connected component

y £ E

[0,1] -> E with

such that there is a continuous,

p(0) = x, p(l) = y, where

[0,1] C R

is the unit interval. We will not prove this result here, although the results of this chapter are sufficient for constructing a proof.

Essentially, one

must look closely at global stratification and local geometry near a point. Note that if our ground field is the field of real numbers, it is not obvious from the considerations leading to 8.13.14 that a semi-algebraically connected set is topologically connected.

However, since

[0,1] C K

is both

topologically and semi-algebraically connected, the fact that components are path components in general, does imply this fact about real numbers.

IE.

Finally we discuss briefly smooth semi-algebraic functions.

simplicity, we restrict our attention to open subsets

n

U C R.

For

We begin by

giving a little more structure to the graph of a continuous semi-algebraic function

f:

U -• R.

Proposition 8.13.15. and suppose

P(x,...x ,y)

P(x,f(x)) = 0 ,

x £ U.

(b)

If

P^

f:

U -* R

is continuous, semi-algebraic

is a polynomial of least degree in

Let

irreducible factors in (a) The

Suppose

P =

p # #p 1 --- r

be the

are distinct, that is, P

P± = P/Pi

and

P

into

is square free.

U, and

P^CxjfCx)) = 0

P.

vanishing on the graph of

IL

are

if x *= U^.

(P^) C RfX-^.-.X ,Y] are convex prime ideals.

(d) dimension (U - U U\) < dimension(U) = n.

262

factorization of

U\ = {x e uli^U.f(x)) * 0}, then the

(c) The principal ideals

If some

such that

RfX^-.X ,Y], then:

non-empty and disjoint open subsets of

Proof:

y

divided

In particular,

UCUJj..

P, we would find a polynomial of lower degree

f by dividing by

P^.

Similarly, if some

u\ = 0,

then

Pi

and

P.

would vanish on the graph of

have lots of zeros in

R

(P.) .

that the algebraic simple zeros of IJL. of

u\ f

are obviously disjoint, over

U-.

follows now from

8.8.6, since the

Moreover, the other results of 8.8 imply P^

are dense on the graph of

f

over

In particular, no polynomial can vanish on any open subset of the graph f

over

IL

Suppose graph of

f

unless

P^

U - U U. over

V.

divides it.

contained an open set Then

But all

P-

vanish on

dense in

P. U

Let

F C R

denote the

is convex and must have

n, which is therefore principal, say

(Q).

F, and applying the arguments of the paragraph above

Q, we would deduce that all

since the

V.

I(F) CR[X,...X ,Y]

some associated prime of dimension

to

The

obviously must vanish on the graph of

The convexity of the ideals (P.)

f.

P. G (Q). As this is clearly impossible

are distinct irreducible polynomials, we conclude

Uu.

is

and (d) follows.

•

We now want to define the subring entiable semi-algebraic functions

f:

C 1 (U) C C°(U) U -• R.

of continuously differ-

We first assume

f G C°(U),

then assume the limits f (x, . .. ,x.+e,.. .xn) - f (x, .. . x j (3f/3x.)(x) = 1

exist in

R

the

property.

£-6

x G u

for all

lim e - 0

x G u

and define functions

(3f/8x i ):

U -* R

We can apply the chain rule to our relation

with

0 = P(x,f(x)),

of lowest degree and this gives

0 = 3P(x,f(x))/3xi

= (3P/3Xi)(x,£(x)) + ((3P/3y)(x,f(x))((3f/3xi)(x))

Since

(3P/3y)(x,f(x) ^ 0

where

P = P^-.P

is dense in

U.

U, and because no

as in 8.13.14, we deduce that The function

since it can be written 8.13.11,

on

3f/3x i:

U -• R

Sf/Sx^

Pi

divides

3P/3y

V = { x G u | (3P/3y) (x,f (x)) i 0}

is obviously semi-algebraic over

(-(3P/3xi)/(3P/3y))(x,f(x)) also belongs to

if

x G V.

Thus by

C°(U).

263

V,

We refer to such functions The set of such

C (U) C C°(U)

U -* R

as

is a subring.

r

by requiring that r,

f

C -semi-algebraic functions. By iterating the procedure,

f G C r (U) C C ^ ^ U ) C ••• C C°(U),

C -semi-algebraic functions

we can define

including

f:

have continuous partial derivatives of order up to and

1 < r < °° .

hence we have a sheaf of

Clearly,

Cr-functions are finitely collatable,

Cr-functions, defined on (variable) open subsets

U C Rn. There is no difficulty whatever now in extending the inverse function theorem, Proposition 8.7.1, to the case of a (Rn,0) -* (Rn,0) injectivity of

C^-map

with non-singular derivative Y

Y = (Y....Y ) :

( O Y - ^ X . ) (0)) .

The local

is proved by standard estimate arguments using the

hypothesis of differentiability.

The local surjectivity of

Y

is proved just

as in the earlier proof of 8.7.1, using the minimum value property of continuous semi-algebraic functions on closed, bounded sets. In fact, we will sketch another proof, similar to a standard proof in the classical case of real numbers.

Beginning with any

x 1 ...x n

y, ...y

near

0, and

C -functions

(3y./9x.)(O) = id, one shows that

of the

y.. ,x_,... ,x

C -coordinate system

x., with

is a

C

y(0) = 0

and

coordinate system.

This argument uses "completeness" in the form of the intermediate value property on intervals for continuous functions.

Since an interval is

connected (in our sense), we have the intermediate value property from 8.13.10 at our disposal.

The inverse function theorem is then proved by

iterating this substitution procedure. Note that even if originally the functions and the the

y"2»»«»»y

will be

y\

x^

are the standard coordinate

are polynomials, at the second step of this proof,

will not generally be polynomials in

C -semi-algebraic functions of

y, ,X2»**«>x , but

y,,x~,...,x . Thus, this proof

would not have been feasible in the special case dealt with in 8.7. The

C°° theory is quite different from the

example, if

U C R

is open, r < °° , then

Specific examples are easy. if

x _> 0

x £ 0.

264

and

Then

f(x) = 0 f-g = 0.

if

Let

Cr

C r (U)

f,g G C ((-1,+1))

x £ 0,

g(x) = 0

if

theory, 1 £ r < °° .

For

is not an integral domain. be defined by x >_ 0

and

f(x) = x

g(x) = x 4

if

If U

is connected, then

if U = U U.

U

is an integral domain.

is the decomposition of

U

r

components, then obviously on

C (U)

(In general,

into disjoint open connected

r

C (U) = IT C (U.)»

0 £ r £ ° ° .) The C°°-functions

are known as Nash functions and have been widely studied in the case

of ordinary real numbers, [30] through [42]. following.

We use the notation n

algebraic functions at

x £ R ,

Proposition 8.13.16.

Cx

for the ring of germs of

C°° =

lim

P(x,f(x)) s o ,

(b) The function C (U) "* C

U

a connected, open,

f £ C (U). Then:

(a) There is an irreducible polynomial such that

C°° semi-

C°°(U).

0 ^ U ^ Rn,

Suppose

semi-algebraic set, and suppose

The basic result here is the

P(X 1...X n ,Y) G R ^ . - . X ^ Y ]

x G U.

f

is determined by its germ

[f] E C~, that is,

is injective.

(c) The germ

{f} =

2 K.—U

[f] £ C^

2 2JX

is determined by its formal power series

(i ) !

(i ) !

-L

Tl

• ~"iC

3

i 1

that is, C* "*-R[[X . ,.Xn]] (d) The power series

in (°)X1

'"

X

R

n

tE x i•' t X n ] ] >

C/A._ • • • O-A. n

is injective. {f}

is a formal solution of the equation

P ( x r . . X n , {£}) = 0. (e) Given an irreducible polynomial series

{f}

PCX^.-X ,Y) and a formal power

P(X 1 ...X n , {£}) = 0, then there is a germ

such that

[f] G C~

with underlying power series {£}.

Sketch of Proof:

Let

P(x,f(x)) = 0 ,

P = P^-.P

8.13.15 into distinct irreducible factors, UL C u P/P.

is non-zero.

factored as in

the open subset where

To prove (a), we must prove that no

x E U

belongs to

U. n U.,

i ^ j , because it then follows from connectedness that there is

only one

U..

If

We may as well prove

0 G u" , then we can find (xQ,f(x Q)) ^ 0.

Near

O ^ U . HU., x G u,

i^j.

arbitrarily near

(x Q ,f(x o )), the graph of

f

0

such that

coincides with

265

the zeros of

P^

The non-vanishing of

compute all partial derivatives the coefficients of

(3P1/8y) (xQ,f ( X Q ) )

(8 f/8x )(x Q ),

I = (i,...ijj, in terms of

P.^, by simply iterating the chain rule and using the

identity

0 = P^XjfCx)), near

variable

x, then

xQ.

For example, if there is only one

0

= fr tvf<xo» + IT cvf<xo» f R11"1

TTE C R

is semi-

algebraic.

Actually, the proof provides an algorithm for writing down semi-algebraic set, in terms of the polynomials used to define

TTE as a E.

However,

we will leave to the reader the details of keeping track of this algorithm through the various steps of the proof. Proposition A.I can be reformulated in logical language as "elimination of quantifiers".

Specifically, by a polynomial relation

we mean a finite sentence built up from basic relations where

p

is a polynomial with coefficients in

tives "and", "or", "not". such that the sentence algebraic set in the type 3 G

or

R

p(x, ...x ,t.....t ) > 0,

R, using the logical connec-

Thus, the set of points

A(x.....x ,t,...t )

(x_...xn,t,...t ) £ R n + m

is true, is just a general semi-

- B y an elementary sentence, we mean a sentence of

(Q.^) • • • (Qnx n)A(x,.. .xn ,t, .. .t ) , where the

V, and

A

A(x . . .x jt^. . .t )

is a polynomial relation.

R m |(Q 1 x 1 )•••(Q n x n )A(x 1 ...x n ,t 1 ...t m )}.

Qi

Consider the set

are quantifiers, ^C t i--- t m )

Another form of the Tarski-Seidenberg

theorem is that this set is semi-algebraic.

Proposition A. 2.

268

There is a polynomial relation

B(tj...t )

(i-n

an algorithm for producing

B), such that

BC^.. .tm) •(Q 1x 1 ) • • • ( Q ^ ) •

A(x 1 ...x n) t 1 ...tJ. Since

(Vxn)A

is equivalent to

~(3x n )(~A), it is clear by induction

that Proposition A.2 needs to be proved only for is what we actually prove below. m

R |(3x 1 )A(x 1 ,t 1 -..t m )} set

n = 1

In this case, the set

is just the projection to

{(Xpt-L..-tm) E R m +

\k(xlitl.. -t m )}.

R

m

and

Q^ = 3.

This

{(t,...t ) £ of the semi-algebraic

We see therefore that Propositions

A.I and A.2 are equivalent. As an example of the sort of sets described by elementary sentences, we mention the topological closure of a semi-algebraic set E = { ( t r . . t m ) eR m |(Vx o )(3x 1 )-..(3x m )(x o = 0 or m o i

E C Rm,

^...xJGE

and

. y w < *o}In order to prove Proposition A.2, we need to introduce temporarily a new notion.

Consider functions

values in

Note that the set of polynomials in one variable of degree

Sk( ))

=

is

^

Continuing this type of argument, we

{(q,x)|sign QCg-^Cx),.. . ,gk(x)) = A}

is semi-

algebraic, as desired.

£j_(b) < £ 2 (b) < • • •, defined.

+

*•• + b n

be a polynomial with real roots

b = (b Q ...b n ).

Of course, the

£^

are not everywhere

The main result needed to prove Proposition A.2 is the following.

Proposition A.5.

Proof: n = 1.

D

p(x) = b Q x n +b^x 1 '

Let

The

^(b)

are psemi-algebraic functions of

We argue by induction on

Since the derivative

(nbQ,(n-l)b1,...,b

p'(x)

The data

has degree

_< n-1, with coefficients

^), we have by induction that the roots of

p'(x), say

sign(bQ),...,sign(bn),sign pCnjCb)),...,sign p(nk(b))

the number of roots of the intervals

of

p(x)

as well as locating each

and on

This is so because C~°°>Tli)> (iiv*00)*

an(

p(x)

p(x)

270

^(b)

in one

^ because the

at

sign(b-)

determine the

J

± °°. {(q,b) G R d + 1 x R n + 1 1 sign qC^Cb)) = \}

We must show that the sets semi-algebraic.

determines

is monotonic between any two roots

IK behavior of

b = (b Q ...b n ).

(- °°,ri-.(b)), (n- (b) ,n-+ 1 (b)) , (nk(b),°°), or among the points

n-^Cb) ,.. . ,n k(b) . p'(x)

£. (b)

b.

n, the Proposition being trivial if

n-j^b) < n 2 (b) < ••• < Tik(v<j (b), are psemi-algebraic functions of

of

By

{(q,y.....yk 2,x)|sign Q(y,.-.yk 2»

semialgebraic.

eventually conclude that

is a polynomial.

{(q,y1-..yk_2,x)|sign F(a Q ...a d ,y 1 ...y k _ 2 ,g k _ 1 (x),x)

is semi-algebraic, and it follows that g

F

We can divide

q

by

p

and write

are

q(x) = s(x)p(x) +r(x),

where

degree(r) < degree(p).

Then

q(^(b)) = r ^ C b ) ) , hence

{(q,b)|sign qCqCb)) = A} = {(q,b)|sign r(q(b)) = X}. ficients of

r

are polynomials in the coefficients

b =

(b0---bn)

of

r(x)

of

q

and

p.

b

psemi-algebraic functions of

= 0, b, ^ 0, and so on.

(a,b).

signCr^Cb) -Yj(a,b)), sign p(y^(a,b)). a psemi-algebraic function of

E $ = {(a,b) e R sign r(^i(b))

is constant on

(a,b).

x R n+1 |(a,b)

(Strictly speaking, there

contains $

$

E$.

sign(b.), sign p(r|.(b)),

By Lemma A.4, r ^ O ) - Yj(a,b)

is

Therefore, any one of the finitely

$}.

Therefore, the set

is determined by

which interval

^ ( b ) , or whether

We claim that

{(q,b)|sign r(^i(b))=

$, we observe first that

(- °° ,ru (b)), (n- (b) >r\., (b)), or 1 3 j+1

^ ( b ) = n-(b), some

Y-(a,b)

of

r(x)

j.

lie in this interval.

^(b),

or

whether

a b

^ ( b ) = Y-( > )>

(- 00 ,Y 1 ), (Yj>Yj+ 1)>

is constant on the intervals

K

some

3-

Thirdly,

p(Y-(a,b))

(-°° ,Y-i(a>b)), (Y- (a,b) >Y- + 1 C a > b )) > o r

determine the interval

(ru (b),°°)

Secondly, we know from"

is monotonic on this interval, hence the values of sign

which contains

bQ ^ 0

E,, hence is semi-algebraic.

sign rC^Cb))

which of the roots

p(x)

r(x), namely if

realizes sign data

is a finite union of some of the

we know from

y. < • • ' < Y m

$ , for this sign data determines a semi-algebraic set

d+1

To see that

and

This causes no problem.)

Consider the sign data consisting of

many possibilities,

a = (a o>..a,)

Thus, by induction, the roots

are cases to l?e distinguished in the formula for or if

Moreover, the coef-

But

(Ym>°°~) > arid

(Ym(a,b),°°) s

ign

our

r

(x)

claim is

established.

D

Proof of Proposition A.2: Seidenberg theorem. polynomials

It clearly suffices to show that given finitely many

^Cx-j^.-.x^

sign f.(x,...x ) = A.} in

Xj

signs \^3 the set

and

is semi-algebraic.

(x 2 ...x n ).

f^, say

^•(x2-•-xn)»

Consider the sign data

^ i -, say

£^

Alternatively,

=

\

It

depends only on

,£ j+ 1 ) , sign

f^,-).

is semi-algebraic.

•

We can now use elimination of quantifiers to eliminate the notion of psemi-algebraic functions.

Proposition A.6.

A function

if and only if the graph of

Proof: function set.

f

f,

f:

X -> R,

X C R m , is psemi-algebraic

F C R m x R, i s a semi-algebraic set.

A very special case of the definition of a psemi-algebraic shows that the set

But this is Conversely, if

polynomial relation {(q>x)I(3t)(A(x,t) Proposition A.2.

272

This observation is due to Efroymson [36].

{(t,x)|f(x) -t = 0}

is a semi-algebraic

F. F

is semi-algebraic, let

A(x,t). and

Then

F = {(x,t)|A(x,t)}

for some

{(q,x)|sign q(f(x)) = X} =

sign q(t) = X]

which is semi-algebraic by •

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277

Notation

(Introduction not included)

Page 32 32 32 33 35 35 36

Page Spec(A,?)

106

m

S ec fA 13 pec t >+Z(X) D(a)

106

r(u, 0 )

116

T>(u, ^ x )

116

106 106

36 36 a,+ '

38

a

42, 81

u{ gi>

162

45

w{ gi >

162

H(X)

46 ^

^&J

162

AH(X) GA

53

Fg

54

-

(I:J)

54

163 163

60

X(A,P)

66

(A(S), P(

77

166 173 X(B,15)d 197

500

83

T

R(A,13)

84 85

vV^gjL^ {g }

i(n)

ftQ

89,Qc; 95

i

^

233

97 98

233

^P ( ^"")

99 (A,£ A )

®

(B/P B )

102 102

233

233 235

C°(U) C°(U) r

259

C (U)

264

C°°(U) C"(U)

264

103 T~

* * 104

278

Index

(Introduction not included)

absolute hull, 53, 54 affine coordinate ring, 173, 201, 219 affine order, 41 Archimedean closed subfield, 140 Archimedean extension, 138, 139 Artin, E., 42, 100, 122, 130, 131, 187 188, 196 associated primes, 59, 60, 61, 73, 95, 96 bounded filter, 235 center of a filter, 235 closed semi-algebraic set, 163, 172, 201 closed, semi-algebraic set, 164, 215, 219, 220, 258 closure of a set, 163 codimension, 212, 213 Cohen, P., 42, 165, 268 concave multiplicative set, 83, 93 connected semi-algebraic set, 249, 257, 262 continuous sections of a sheaf, 115, 116, 117 continuous semi-algebraic function, 252, 253, 255, 256, 257 contraction of an order, 35 convex set, 33 convex hull, 46, 47 Dedekind cuts, 137, 138 degenerate points, 197 derived order, 39, 42, 64, 81, 97, 98, 188, 199, 200 derived set, 36, 37, 38, 39 dimension, 212, 219 direct limit, 103, 105 Dubois, D., 187 Efroymson, G., 216, 227, 272 elementary sentence, 268 extension of an order, 35, 40, 120, 121, 122, 130 Fat City, 219 fibre product, 101 fibre sum, 102, 104, 105 filter, 43, 233 filter at infinity, 235 formally real field, 34, 149

Hilbert, D., 189 Hilbert 17th problem, 42, 188 ideal absolutely convex, 52, 53, 54, 55, 88, 89, 156 convex, 45, 46, 52, 81 maximal convex, 49, 50, 63, 87, 93 167, 185 minimal prime, 58, 60, 91, 92, 108 primary convex, 57, 64, 74, 75, 82, 89, 94 prime convex, 49, 51, 63, 66, 74, 90, 93, 127, 186 Jacobson radical, 85, 86, 99 Krull dimension, 224, 225 Krull, F., 139, 149, 150 Krull valuation, 146 Lang, S., 130, 184 localized order, 77, 79, 80, 81, 82, 111, 112, 113 maximal order, 33, 37, 38, 39, 44 morphism, 32 Nash functions, 265 nil radical, 37, 46, 47, 48, 51, 90, 167 Noetherian-Grothendieck topology, 248, 252 non-degenerate points, 197, 198, 199, 201, 202, 215, 218 open semi-algebraic set, 163, 164 open, semi-algebraic set, 164 order, 32 partially ordered ring, 32 partition of unity, 44 polynomial relation, 268 (POR), 32, 34 (PORCK), 33, 34, 55, 56, 88, 110, 114, 117, 119 (PORNN), 33, 34, 56 (PORPP), 33 prefilter, 233, 234 product order, 44

279

psemi-algebraic function, 269, 270 272 quotient order, 45, 52, 56, 58, 81, 201, 202 rank, 213 real closed field, 130, 131, 132, 134 135 refinement of an order, 33, 80, 81 96, 97, 150, 151, 157, 189 RHJ algebra, 167, 168, 169, 170, 171, 199, 200, 201, 202, 219 Risler, J. J., 187 Schreier, 0., 122, 130, 131 semi-algebraic set, 162, 268 semi-field, 49, 50, 85 semi-integral extension, 126, 127 154, 155, 170 semi-Noetherian ring, 57, 58, 95, 96 shadow of a set, 83, 84 shadow of 1, 84, 88 sheaf of partially ordered rings, 41, 112, 249 signed place, 145, 148, 161, 184, 186 190, 206, 228, 242 simple point, 213, 215 simple refinement, 36, 37 simplicial complex, 43, 69 split valuation, 147

280

Stengle, G., 100, 173, 189, 193, 195, 196 strong 'j^chain, 225, 226, 228, 231 strong topology, 247 structure sheaf rational, 248, 250 semi-algebraic, 248, 252, 259 smooth, 248, 262, 265 Zariski, 112, 115, 116, 249 Sturm algorithm, 135, 136 symbolic powers, 89, 95 Tarski-Seidenberg theorem, 163, 164, 165, 203, 206, 207, 215, 233, 258, 268, 271 total order, 37, 64, 141, 143, 145, 156, 178, 234, 238 ultrafilter, 233, 234 valuation, 146, 242 valuation ring, 139, 149, 152 weak order, 33, 168 weak $-chain, 224, 227 Whitney, H., 261 Zariski dense set, 43, 238 Zariski, 0., 244 Zariski topology, 106, 107, 108, 109, 110 zero divisors, 42, 60, 91, 92