Lectures on Algebra

S S Abhyankar

Lectures on Algebra Volume I

Lectures on Algebra Volume I

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S S Abhyankar Purdue University, USA

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Lectures on Algebra Volume I

Y | ^ World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI

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LECTURES ON ALGEBRA Volume I Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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PREFACE

In universities and colleges it has become customary to give two algebra courses, the first being called abstract algebra and the second linear algebra. The present volume, Lectures on Algebra I, is meant as a text-book for an abstract algebra course, while the forthcoming sequel, Lectures on Algebra II, should serve as a text-book for a linear algebra course. The author's fondness for algebraic geometry shows up in both volumes, and his recent preoccupation with the application of group theory to the calculation of Galois groups is evident in the second volume which contains more local rings and more algebraic geometry. Both volumes are based on the author's lectures at Purdue University during the last several years. An attempt has been made to make these volumes self-contained. The reader may prefer to start with the sixth lecture which gives a rapid summary of the first five lectures. He may also find it helpful to look at the detailed contents printed at the end of the volume just before the index; this is particularly significant for the enormous (about 300 pages) Section §5 of Lecture L5. When in a certain lecture we are referring to an item from another lecture, the citation of the other lecture precedes the citation of the item. Thus, for instance, in the proof of Theorem (Q4)(T13) in Lecture L5, the reference L4§5(011) is to Observation (Oil) of §5 of Lecture L4, whereas the reference ( T i l ) is to Theorem ( T i l ) of Lecture L5. Frequently, assertions made in one place are proved or expanded later on. For this purpose, forward reference is indicated by [cf.]. For instance, on page 4, at the end of the sentence "This is unique up to isomorphism, i.e., between two copies of it there is a one-to-one onto map preserving sums and products;" the phrase "[cf. L5§5(Q32)(T138.2)]" means that the proof of the preceding claim that there is a unique field GF(q) of q elements, will be given in Theorem (T138.2) of Quest (Q32) of Section §5 of Lecture L5. Like a Russian Petrushka doll, there are many books within this book. For instance, the first three lectures, LI, L2, L3 constitute a booklet on a basic abstract algebra course. The sixth lecture L6 by itself constitutes another such course. These two alternatives togther make up a larger such booklet. The fourth lecture L4 is a booklet on commutative algebra. It is continued in the fifth lecture L5 which may be V

PREFACE

VI

viewed as a treatise on commutative algebra. Within it, Quest (Q31) is a pamphlet on Suslin's work on projective modules and special linear groups over multivariable polynomial rings. Finally, Sections §§2-5 of L3, §§8-9 of L4, and Quests (Q33)-(Q35) of L5§5 form a short course on algebraic geometry. Thanks are due to Sudhir Ghorpade, Nan Gu, Nick Inglis, Valeria Grant Perez, Avinash Sathaye, David Shannon, Balwant Singh, Umud Yalcin, and Ikkwon Yie for much help. Thanks are also due to NSF Grant DMS 99-88166 and NSA Grant MSP H98230-05-1-0040 for financial support. Shreeram S. Abhyankar, Mathematics Department, Purdue University, West Lafayette, IN 47907, USA, e-mail: [email protected]

CONTENTS

Lecture LI: QUADRATIC EQUATIONS §1: Word Problems §2: Sets and Maps §3: Groups and Fields §4: Rings and Ideals §5: Modules and Vector Spaces §6: Polynomials and Rational Functions §7: Euclidean Domains and Principal Ideal Domains §8: Root Fields and Splitting Fields §9: Advice to the Reader §10: Definitions and Remarks §11: Examples and Exercises §12: Notes §13: Concluding Note

1 1 3 3 6 8 9 13 14 16 16 23 27 29

Lecture L2: CURVES AND SURFACES §1: Multivariable Word Problems §2: Power Series and Meromorphic Series §3: Valuations §4: Advice to the Reader §5: Zorn's Lemma and Well Ordering §6: Utilitarian Summary §7: Definitions and Exercises §8: Notes §9: Concluding Note

30 30 34 39 43 44 52 52 59 60

Lecture L3: TANGENTS AND POLARS §1: Simple Groups §2: Quadrics §3: Hypersurfaces §4: Homogeneous Coordinates

61 61 63 64 66

viii

CONTENTS

§5: Singularities §6: Hensel's Lemma and Newton's Theorem §7: Integral Dependence §8: Unique Factorization Domains §9: Remarks §10: Advice to the Reader §11: Hensel and Weierstrass §12: Definitions and Exercises §13: Notes §14: Concluding Note

70 72 77 81 82 83 83 90 98 98

Lecture L4: VARIETIES AND MODELS §1: Resultants and Discriminants §2: Varieties §3: Noetherian Rings §4: Advice to the Reader §5: Ideals and Modules §6: Primary Decomposition §6.1: Primary Decomposition for Modules §7: Localization §7.1: Localization at a Prime Ideal §8: Afnne Varieties §8.1: Spectral Afnne Space §8.2: Modelic Spec and Modelic Affine Space §8.3: Simple Points and Regular Local Rings §9: Models §9.1: Modelic Proj and Modelic Projective Space §9.2: Modelic Blowup §9.3: Blowup of Singularities §10: Examples and Exercises §11: Problems §12: Remarks §13: Definitions and Exercises §14: Notes §15: Concluding Note

100 100 104 105 107 108 134 136 137 144 146 152 152 153 154 157 159 160 161 171 172 195 200 201

Lecture L5: PROJECTIVE VARIETIES §1: Direct Sums of Modules §2: Grades Rings and Homogeneous Ideals §3: Ideal Theory in Graded Rings §4: Advice to the Reader §5: More about Ideals and Modules

202 202 206 209 216 216

CONTENTS

(Ql) Nilpotents and Zerodivisors in Noetherian Rings (Q2) Faithful Modules and Noetherian Conditions (Q3) Jacobson Radical, Zariski Ring, and Nakayama Lemma (Q4) Krull Intersection Theorem and Artin-Rees Lemma (Q5) Nagata's Principle of Idealization (Q6) Cohen's and Eakin's Noetherian Theorems (Q7) Principal Ideal Theorems (Q8) Relative Independence and Analytic Independence (Q9) Going Up and Going Down Theorems (Q10) Normalization Theorem and Regular Polynomials (Qll) Nilradical, Jacobson Spectrum, and Jacobson Ring (Q12) Catenarian Rings and Dimension Formula (Q13) Associated Graded Rings and Leading Ideals (Q14) Completely Normal Domains (Q15) Regular Sequences and Cohen-Macaulay Rings (Q16) Complete Intersections and Gorenstein Rings (Q17) Projective Resolutions of Finite Modules (Q18) Direct Sums of Algebras, Reduced Rings, and PIRs (Q18.1) Orthogonal Idempotents and Ideals in a Direct Sum (Q18.2) Localizations of Direct Sums (Q18.3) Comaximal Ideals and Ideal Theoretic Direct Sums (Q18.4) SPIRs = Special Principal Ideal Rings (Q19) Invertible Ideals, Conditions for Normality, and DVRs (Q20) Dedekind Domains and Chinese Remainder Theorem (Q21) Real Ranks of Valuations and Segment Completions (Q22) Specializations and Compositions of Valuations (Q23) UFD Property of Regular Local Domains (Q24) Graded Modules and Hilbert Polynomials (Q25) Hilbert Polynomial of a Hypersurfaces (Q26) Homogeneous Submodules of Graded Modules (Q27) Homogeneous Normalization (Q28) Alternating Sum of Lengths (Q29) Linear Disjointness and Intersection of Varieties (Q30) Syzygies and Homogeneous Resolutions (Q31) Projective Modules Over Polynomial Rings (Q32) Separable Extensions and Primitive Elements (Q33) Restricted Domains and Projective Normalization (Q34) Basic Projective Algebraic Geometry (Q34.1) Projective Spectrum (Q34.2) Homogeneous Localization (Q34.3) Varieties in Projective Space (Q34.4) Projective Decomposition of Ideals and Varieties

ix

216 218 219 220 225 229 230 236 241 247 261 268 272 277 280 300 311 340 341 344 345 348 354 364 372 381 385 393 397 399 401 408 414 433 441 514 529 534 534 536 541 545

x

CONTENTS

(Q34.5) Modelic and Spectral Projective Spaces (Q34.6) Relation between AfRne and Projective Varieties (Q35) Simplifying Singularities by Blowups (Q35.1) Hypersurface Singularities (Q35.2) Blowing-up Primary Ideals (Q35.3) Residual Properties and Coefficient Sets (Q35.4) Geometrically Blowing-up Simple Centers (Q35.5) Algebraically Blowing-up Simple Centers (Q35.6) Dominating Modelic Blowup (Q35.7) Normal Crossings, Equimultiple Locus, and Simple Points (Q35.8) Quadratic and Monoidal Transformations (Q35.9) Regular Local Rings §6: Definitions and Exercises §7: Notes §8: Concluding Note

547 548 552 552 553 555 555 559 566 567 569 577 578 596 597

Lecture L6: PAUSE AND REFRESH §1: Summary of Lecture LI on Quadratic Equations §2: Summary of Lecture L2 on Curves and Surfaces §3: Summary of Lecture L3 on Tangents and Polars §4: Summary of Lecture L4 on Varieties and Models §5: Summary of Lecture L5 on Projective Varieties §6: Definitions and Exercises

598 598 603 606 608 611 634

BIBLIOGRAPHY

689

DETAILED CONTENT

691

NOTATION-SYMBOLS

713

NOTATION-WORDS

717

INDEX

725

Lecture LI: Quadratic Equations §1: W O R D PROBLEMS Consider the following word problem. One morning I went to the garden and plucked some roses. Seeing that there were not enough I went to as many flower shops as I had roses and from each of them I purchased as many roses as I had plucked. Thus armed with sufficient flowers I went to the Ganesh Temple, Ganesh being the God of Learning and especially Mathematics. I put at his feet four times as many roses as I had plucked. Then I went to the Shiva Temple and deposited ten roses. The remaining eight roses I took to my spouse. How many roses did I pluck? Now in algebra what we do not know we call x. So having plucked x roses, from the various shops I bought x x x = x2 roses. Thus armed with x2 + x roses I offered to Ganesh Ax roses and then having deposited ten to Shiva the eight for my spouse makes x2 + x = Ax + 10 + 8. Bringing everything to one side gives us the quadratic equation x2 - 3x - 18 = 0. We can solve this quadratic equation by completing the square. In other words we want to add a quantity to x2 — 3x to make it a complete square. By the binomial theorem we have (x + y)2 = x2 + 2xy + y2. So we want 2y = —3, i.e., y = =£ and hence y2 = | . Thus transferring 18 to the right hand side and then adding | to both sides we get x2 - 2.x + | = f + 18, i.e., (x - | ) 2 = f + 18 = ^p- = f = ( | ) 2 . Therefore x - § = ± § . Thus x = | + | = 6 o r a ; = | - | = - 3 . Discarding the negative solution —3 as not applicable in this case we conclude that I plucked 6 roses. This completing the square method of solving a quadratic equation was conceived around 500 A.D. by the Indian mathematician Shreedharacharya. It was put in verse form in 1150 A.D. by another Indian mathematician Bhaskaracharya in his book [Bha] on Algebra called Beejganit. Thus, writing Y for the unknown, we have the quadratic equation aY2 + bY + c where a, b, c are its coefficients with a ^ 0. To make it monic, i.e., to arrange the coefficient of the highest degree term to be 1, we divide by a to get Y2 + BY + C with B = | and C = ^. Now we complete the square by writing R\2

/

Y

2

+ BY + C=lY+-j 1

R2

+C-—.

2

LECTURE LI: QUADRATIC EQUATIONS

Putting back the values of B and C, we get the solutions _ — b ± y/b2 — 4ac ~ 2o ' In the above word problem, first I thought of bringing 5 flowers to my spouse. That would make a = 1, b = —3, c = - 1 5 and putting this in the above formula we would get the solutions to be 3±%^. Since 69 has no integer square root I would be faced with the dilemma of picking an irrational number of flowers. That is why I decided to bring 8 flowers to my spouse since I knew this would lead to a constant term in my equation which could be factored into two factors whose negative sum equals the middle term. What I am referring to is the identity (Y - a){Y -/3) = Y2-(a

+ 0)Y + aP = Y2 + BY + C

which tells us that C — a0 and B = —(a + 0), i.e., the constant term is the product of the roots and the coefficient of Y is the negative sum of the roots. Around 1500 A.D., similar but much more complicated formulas for solving cubic and quartic equations by radicals were given by Cardano and Ferrari in Italy. Here by radicals we mean successively extracting square roots, cube roots, and so on. Then for the next three hundred years people tried to solve general quintic equations, where by solving they meant solving by radicals. This was proved to be impossible, first by the Norwegian mathematician Abel in 1820 and then by the young French mathematician Galois in 1830 [Gal]. But Galois went much further and gave a criterion of solvability for any polynomial equation of any degree. What he did was to associate to the equation a certain finite permutation group, which is now called the Galois group, and to prove that the equation can be solved by radicals if and only if its Galois group is solvable in a technical sense which he introduced. To explain this let f(Y)=a0Yn

+ aiYn~1+---

+ an

with

a0 ± 0

be a polynomial of any degree n. The coefficients ao, a i , . . . , an could be rational numbers, or real numbers, or complex numbers. All these are examples of fields, i.e., collections of objects in which we can carry out the operations of addition, subtraction, multiplication and division; in the last case the quantity we are dividing by is required to be nonzero. Letting the coefficients ao, a\,. • •, an of / belong to any field K, we find its roots a i , . . . , an in some bigger field giving us f(Y) =

a0(Y-a1)...(Y-an).

We assume the polynomial / to be separable by which we mean that the roots ai,...,an are all distinct. The Galois group of / is going to be a certain group of permutations of the roots ai,... ,an. A permutation on a set S is a bijection, i.e., a one-to-one onto map, of S onto itself. Under composition the set of all bijections of

§2: SETS AND MAPS

3

S forms a group which we denote by Sym(S'). In case the set S is finite and its size | 5 | (= the number of elements in it) is n, it is customary to write Sn for Sym(S'). For the size of an infinite set S we put \S\ — oo. Let us now formally introduce the terms set, map, composition, bijection, group, and field. §2: SETS A N D M A P S A map <j> '• S —> T from a set 5 (= a collection of objects called its elements) to a set T is a rule which to every x € S, i.e., element x of S, associates 4>{x) £ T, which is called the image of x under <j>\ this is sometimes indicated by writing x — i > {x) which is read as x maps to (x). The map

(x) ^ (y). The map is surjective (= onto) if for every y £T there is some x £ S with y = cj>(x). The map <j> is bijective if it is injective as well as surjective. A bijection (resp: an injection, a surjection) is a bijective (resp: injective, surjective) map. The composition of maps <j> : S —> T and ip : T —> £/ is the map ^ 0 : S —» C/ given by (tp T the inverse -1 : T —> 5 is defined by <j)~1{y) = x <j)(x) = y. The logical symbols A => B, A