Lectures on Algebra Volume 1

S. S. Abhyankar9812568263, 9789812568267

This book is a timely survey of much of the algebra developed during the last several centuries including its applications to algebraic geometry and its potential use in geometric modeling. The present volume makes an ideal textbook for an abstract algebra course, while the forthcoming sequel, “Lectures on Algebra II”, will serve as a textbook for a linear algebra course. The author’s fondness for algebraic geometry shows up in both volumes, and his recent preoccupation with the applications 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 books are based on the author’s lectures at Purdue University over the last few years.

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Table of contents :
CONTENTS……Page 8
1: Word Problems……Page 12
3: Groups and Fields……Page 14
4: Rings and Ideals……Page 17
5: Modules and Vector Spaces……Page 19
6: Polynomials and Rational Functions……Page 20
7: Euclidean Domains and Principal Ideal Domains……Page 24
8: Root Fields and Splitting Fields……Page 25
10: Definitions and Remarks……Page 27
11: Examples and Exercises……Page 34
12: Notes……Page 38
13: Concluding Note……Page 40
1: Multivariable Word Problems……Page 41
2: Power Series and Meromorphic Series……Page 45
3: Valuations……Page 50
4: Advice to the Reader……Page 54
5: Zorn’s Lemma and Well Ordering……Page 55
7: Definitions and Exercises……Page 63
8: Notes……Page 70
9: Concluding Note……Page 71
1: Simple Groups……Page 72
2: Quadrics……Page 74
3: Hypersurfaces……Page 75
4: Homogeneous Coordinates……Page 77
5: Singularities……Page 81
6: Hensel’s Lemma and Newton’s Theorem……Page 83
7: Integral Dependence……Page 88
8: Unique Factorization Domains……Page 92
9: Remarks……Page 93
11: Hensel and Weierstrass……Page 94
12: Definitions and Exercises……Page 101
14: Concluding Note……Page 109
1: Resultants and Discriminants……Page 111
2: Varieties……Page 115
3: Noetherian Rings……Page 116
4: Advice to the Reader……Page 118
5: Ideals and Modules……Page 119
6: Primary Decomposition……Page 145
6.1: Primary Decomposition for Modules……Page 147
7: Localization……Page 148
7.1: Localization at a Prime Ideal……Page 155
8: Affine Varieties……Page 157
8.2: Modelic Spec and Modelic Affine Space……Page 163
8.3: Simple Points and Regular Local Rings……Page 164
9: Models……Page 165
9.1: Modelic Proj and Modelic Projective Space……Page 168
9.2: Modelic Blowup……Page 170
9.3: Blowup of Singularities……Page 171
10: Examples and Exercises……Page 172
11: Problems……Page 182
12: Remarks……Page 183
13: Definitions and Exercises……Page 206
14: Notes……Page 211
15: Concluding Note……Page 212
1: Direct Sums of Modules……Page 213
2: Graded Rings and Homogeneous Ideals……Page 217
3: Ideal Theory in Graded Rings……Page 220
Ql) Nilpotents and Zerodivisors in Noetherian Rings……Page 227
Q2) Faithful Modules and Noetherian Conditions……Page 229
Q3) Jacobson Radical Zariski Ring and Nakayama Lemma……Page 230
Q4) Krull Intersection Theorem and Artin-Rees Lemma……Page 231
Q5) Nagata’s Principle of Idealization……Page 236
Q6) Cohen’s and Eakin’s Noetherian Theorems……Page 240
Q7) Principal Ideal Theorems……Page 241
Q8) Relative Independence and Analytic Independence……Page 247
Q9) Going Up and Going Down Theorems……Page 252
Q10) Normalization Theorem and Regular Polynomials……Page 258
Qll) Nilradical Jacobson Spectrum and Jacobson Ring……Page 272
Q12) Catenarian Rings and Dimension Formula……Page 279
Q13) Associated Graded Rings and Leading Ideals……Page 283
Q14) Completely Normal Domains……Page 288
Q15) Regular Sequences and Cohen-Macaulay Rings……Page 291
Q16) Complete Intersections and Gorenstein Rings……Page 311
Q17) Projective Resolutions of Finite Modules……Page 322
Q18) Direct Sums of Algebras Reduced Rings and PIRs……Page 351
Q18.1) Orthogonal Idempotents and Ideals in a Direct Sum……Page 352
Q18.2) Localizations of Direct Sums……Page 355
Q18.3) Comaximal Ideals and Ideal Theoretic Direct Sums……Page 356
Q18.4) SPIRs = Special Principal Ideal Rings……Page 359
Q19) Invertible Ideals Conditions for Normality and DVRs……Page 365
Q20) Dedekind Domains and Chinese Remainder Theorem……Page 375
Q21) Real Ranks of Valuations and Segment Completions……Page 383
Q22) Specializations and Compositions of Valuations……Page 392
Q23) UFD Property of Regular Local Domains……Page 396
Q24) Graded Modules and Hilbert Polynomials……Page 404
Q25) Hilbert Polynomial of a Hypersurfaces……Page 408
Q26) Homogeneous Submodules of Graded Modules……Page 410
Q27) Homogeneous Normalization……Page 412
Q28) Alternating Sum of Lengths……Page 419
Q29) Linear Disjointness and Intersection of Varieties……Page 425
Q30) Syzygies and Homogeneous Resolutions……Page 444
Q31) Projective Modules Over Polynomial Rings……Page 452
Q32) Separable Extensions and Primitive Elements……Page 525
Q33) Restricted Domains and Projective Normalization……Page 540
Q34.1) Projective Spectrum……Page 545
Q34.2) Homogeneous Localization……Page 547
Q34.3) Varieties in Projective Space……Page 552
Q34.4) Projective Decomposition of Ideals and Varieties……Page 556
Q34.5) Modelic and Spectral Projective Spaces……Page 558
Q34.6) Relation between AfRne and Projective Varieties……Page 559
Q35.1) Hypersurface Singularities……Page 563
Q35.2) Blowing-up Primary Ideals……Page 564
Q35.4) Geometrically Blowing-up Simple Centers……Page 566
Q35.5) Algebraically Blowing-up Simple Centers……Page 570
Q35.6) Dominating Modelic Blowup……Page 577
Q35.7) Normal Crossings Equimultiple Locus and Resolved Ideals……Page 578
Q35.8) Quadratic and Monoidal Transformations……Page 580
Q35.9) Regular Local Rings……Page 588
6: Definitions and Exercises……Page 589
7: Notes……Page 607
8: Concluding Note……Page 608
1: Summary of Lecture LI on Quadratic Equations……Page 609
2: Summary of Lecture L2 on Curves and Surfaces……Page 614
3: Summary of Lecture L3 on Tangents and Polars……Page 617
4: Summary of Lecture L4 on Varieties and Models……Page 619
5: Summary of Lecture L5 on Projective Varieties……Page 622
6: Definitions and Exercises……Page 645
BIBLIOGRAPHY……Page 700
DETAILED CONTENT……Page 702
NOTATION-SYMBOLS……Page 724
NOTATION-WORDS……Page 728
INDEX……Page 736