Commutative Algebra with a View Toward Algebraic G

文件名称: Commutative Algebra with a View Toward Algebraic Geometry(GTM 150)

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详细说明：在抽象代数中,交换代数旨在探讨交换环及其理想,以及交换环上的模Contents Introduction for the bes r 2 requisites 6 a First Cot ourse howled e 0 Elementary Definitions 11 0. 1 Ringy and Ides 0.2 Unique Factorization 0 3 Modules Basic constructions 19 1 Roots of c tative Algebra 21 1.1 Number T 21 1.2 Algebraic Curves and Function Theory 1 8 Invariant Th 1.4 The Basis Theorem 1.4.1 Finite generation of invariants Contents 1.6 Algebra and Geometry: The Nullstellensatz 1. 7 Geometric Invariant Th 1. 8 Projective Varieties 1.9 Hilbert Functions and polynomials 1.10 Free Resolutions and the Syzygy Theorem 44 111 Exercises Noetherian Rings and Modules An Analysis of Hilberts Finiteness Argument Some rings of invariants alg Graded Rings and Projective Geometry Hilb - Spec, and the Zariski Topolo 2 2⊥ Fracti 22H 2 The construction of primes 24 78 26E 7.graded Ring's and Their Localizations Partitions of Unity Glu Idempotents, Products, and Connected Componente 85 3 Associated Primes and Primary Decomposition 87 3.1 Associated Primes 3.2 Prime avoidance 90 3. 9 Prirnary Decomposition 3.4P 3.5 Primary Decomposition in the Graded Case 99 8.6 Extracting Information from Primary Decomposition 8.7 Why Prirnary Decomposition Is Not Unique 102 3.8 tric Interpretati f上 3.9 Symbolic Powers and Functions Vanishing to High Order 8.9.1 A Determinantal Example 3.10飞 xerces General Graded Primary Decompo sition Primary Decomposition of Monomial Ideals The Question of Uniqueness 111 Determinantal Ideals 112 Contents ix Total Quotients 113 Prime Avoidance 113 4 Integral Dependence and the Nullstellensatz The Cayley-Hamilton Theorem and Nakayama's Lemma 119 4.2 Normal domains and the normalization process 125 3 Normalization in the Analytic C 128 4. 4 Primes in an Integral Extension 129 4.5 The nullstellensatz 4.6Ex Nakayama Lcmma 135 Projective Modules and Locally Free Modules Integral Closure of Ideala normalization 137 Normalization and Convexity Nullstellensatz Three More proofs of the nullstellensatz 142 5 Filtrations and the Artin-Rees lemma I45 1 Associated Graded Rings and Module 46 5.2 The Blowup algebra 148 5.3 The Krull Intersection Theorem 150 5. 4 The Tangent Cone 5.5 Exercises 151 6 Flat f allies 155 6.1 Elementary Examples 6.2 Introduction to Tor 159 6.3 Criteria for flatness 6.4 The local criterion for flatness 166 6.5 The Rees algebra 170 Flat Families of graded modul 175 Embedded first-Order De 175 7 Completions and llensel's Lemma L79 7.1 ExaMples and Definitions 179 7.2 The Utility of Completions 7.3 Lifting Idempotents 7.4 Cohen Structure Theory and Coefficient Fields 7.5 Basic Properties of Completi 7.6 Maps from Power Series Ring 198 Modules Whose Completions Are Isomorphic 203 all Topology and Cauchy Sequences 204 205 Contents Coefficient fields 205 Other versions of hensels lemma 206 II Dimension Theory 211 8 Introduction to dimension theory 213 8.1 Axioms for dimension 218 8.2 Other Characterizations of dimension 220 8.2.1 Affine Rings and Noether Normalization 221 8.2.2 Systems of Parameters and Krulls Principal Ideal Theorem 222 8.2.3 The Degree of the hilbert polynomial 223 9 Fundamental Definitions of Dimension Theory 225 9.1 Dimension Zero 227 9. 2 Exercises 22 10 The Principal idcal Thcorcm and Systcms of Parameters 231 10.1 Systems of Parameters and Parameter Ideals 234 10.2 Dimension of Base and Fiber 236 10.3 Regular Local Rings 240 10.4 Exercises ,,242 Determinantal Ideals Hilbert Series of a Graded Module 245 11 Dimension and codimension one 247 11.1 Discrete Valuation Rings 247 11.2 Normal Rings and Serre 's criterion 249 11. 3 Invertible Module 11.4 Unique Factorization of Codimension-One Ideals..... 256 115D d multiplicity 259 11.6 Multiplicity of Principal Ideals 261 11. Exercises 264 Valuation ri 264 The grothendieck Ring 26 1.2 Dimension and Hilbert-Samuel Polynomials 12. 1 Hilbert-Samuel Functions 272 12.2 Exercises 275 Analytic Spread and the Fiber of a blowup 276 Multiplicities Hilbert series 280 Contents xi 1 3 The Dimension of Affine rings 281 3.1 Noether Normalization 281 13. 2 The Nullstellensatz 292 13. 3 Finiteness of the Integral Closure 1 3 4 Exercises 296 Quotients by Finite Groups 296 in Polynomial Rings Dimension in the graded Case 297 Noether Normalization in the Complete Case 298 Products and Reduction to the Diagonal 299 equational Characterization of Systems of Parameters 301 11 Elimination Theory, Generic Freeness, and the Dimension of fiberg 808 14.1 Elimination Thcory 303 14.2 Ge Freeness 307 14 The dimension of fibers 308 14.4 exercises 14 Elimination The 314 15 Grobner Bases 317 ConstriCtive Module Theory 318 Eimination Theory 318 15.1 Mo als and Te 15.1.1 Hilbert Function and polynomial 820 15. 1.2 Syzygies of Monomial Submodules 922 15.2 Monomial Orders 323 15.8 The Division Algorithm 15.4 Grobner Bases .881 15.5 Sy zyg 334 15.6 History of Grobner Bases 15. 7 A Propcrty o crsc Lexicographic Ordcr 338 15.8 Grobner bases Flat Families 312 15.9 Generic initial ideals ,348 15.9.1 Existence of the generic Initial Ideal 349 15.9.2 The Generic Initial Ideal 165.9.3 The Nature of borel-fixed ide 852 15. 10 Applications 355 15.10.1 Ideal Membership 15.10.2 Hilbert Function and Polynomial 355 15.10.3 Associated Graded Ring 356 15.10.4 Eiiminatio 15. 10.5 Projective Closure and Ideal at Infinity 859 15.10.6 Saturation 860 ntents 15.10. 7 Lifting Homomorphisms 360 15. 10.8 Syzygies and Constructive Module theor 361 15.10.9 What's Left? 363 15.11 Exercises 365 15.12 Appendix: Some Computer Algebra Projects..,.,.375 Project 1. Zero-Dimensional Gorenstein Ideals.... 376 Project 2. Factoring Out a general element from an sth Syzygy Project 3. Resolutions over ilypersurfaces ..377 Project 4. Rational Curves of Degree ]+l in Pr 378 Project 5. Regularity of Rational Curve 378 Project 6. Some Monomial Curve Singularities .379 Project 7. Some Interesting Prime Ideals 379 6 Modules of differentials 16. 1 Computation of Differentiala 16.2 Differentials and the Cotangent Bundle 16. 3 Colimits and Localization 391 16.4 Tangent Vector Fields and Infinitesimal Morphisms... 396 6.5 Differentials and Field Extensions 397 16.6a for regularity 401 16.7 Smoo thness and generic smoo thess 404 6.8 Appendix: Another Construction of Kahler Differentials 169 Exercises.,· 409 III Homological Methods 417 17 Regular Sequences and the Koszul Complex 419 17.1 Koszul Complexes of lengths 1 and 2 17.2 Koszul Complexes in General 423 17.3 Building the Koszul Complex from Parts 427 432 17.5 The Koszul Complex and the Cotangent Bundle of Projective Space 435 17. 6 Exer Free Resolutions of Monomial ideals 439 Conormal Sequence of a Complete Intersection 440 Regular Sequences are Like Sequences of Variables 4 40 Blowup algebra and Normal cone of a regular quenc Geometric Contexts of the Koszul complex 412 Contents xiii 18 Depth, Codimension, and Cohen-Macaulay rings 18.1 Depth 447 18.1.1 Depth and the Vanishing ofExt 449 182C Macau 451 18. 3 Proving Primcncss with Scrrc's Critcrion 457 84F nd Depth 460 18.5 Some Examples 462 18.6 Exercises 9 Homological Theory of Regular Local Rings 19.1 Projective Dimension and Minimal Resolutions 469 19.2 Global Dimension and the Syzygy Theorem 474 19.3 Depth and Projective Dimension The Auslander-Buchsbaum Formula 19.4 Stably Frcc Modules and Factoriality of Regular Local Rings 19.5 Exercises 483 Regular Ri 484 The Auslander-Buchsbaum Formula Modules over a dedekind dor 484 48 Projective Dimension and Cohen-Macaulay Rings 485 Hilbert Function and Grothendieck Group The Chern Polynomial 20 Free Resolutions and Fitting Invariant 20. 1 The Uniqueness of Frcc Resolutions ,....,...... 490 20.2 Fitting Ideals 20.3 What Makes a Complex Exact? 20.4 The Hilbert-Burch Theorem 501 20.4.1 Cubic Surfaces and Sextuples of Points in the Plane 53 20.5 Castelnuovo-Mumford Regularity 20.5. 1 Regularity and Hyperplane Sections 508 20.5.2 Regularity of Generic Initial Ideals..,......509 20.5.3 Historical Notes on Regularity 509 20.6卫 Mercies 510 Fitting Ideals and the Structure of Modules Projectives of Constant Rank Castelnuovo-Mumford Regularity 516 21 Duality, Canonical Modules, and Gorenstein Rings 21.1 Duality for Modules of Finite Length 520 21.2 Zero-Dimensional Gorenstein Rings 525 21.3 Canonical Modules and Gorenstein Rings in Higher Dimension 528 XlV Contents 21.4 Maximal Cohen-Macaulay Modules 21.5 Modules of Finite Injective Dimension 530 21.6 Uniqueness and (Often) Existence ..534 21.7 Localization and Completion of the Canonical Module . 536 21.8 Complete Intersections and Other Gorenstein Rings.,.. 587 21.9 Duality for Maximal Cohen-Macaulay Modules 538 21.10 Linkage ,,,,,,,.539 21.11 Duality in the Graded Case 545 21 12 Exercises ,546 The Zero-D onal cas d duali 546 ner 548 The Canonical module as ideal Linkage and the Cayley-Bacharach Theorem 552 Appendix 1 Field Theory 555 Al.1 Transcendence Degree 55: Al.2 Separability 55 Al.3 D-B 559 Al. 3. 1 Exercises 562 ppendix 2 Multilinear algebra a2.1 Introductie A2.2 Tensor Pro ducts 567 A2.3 Symmetric and Exterior Algebras 569 a2.3.1 Bases 572 a2.3.2 Exercises 4 A2.4 Coalgebra Structures and Divided Powers 575 A2.4.1 S( F and SMas Modules over One another 582 A2.5 Schur Functors 84 A 2.5.1 Exercises 587 A2.6 Complexes Constructed by Multilinear algebra 58 A2.6. 1 Strands of the Koszul Complex 591 A2 6.2 Exercises 603 Appen dix 3 Homological Algebra 611 A3.1 Introduction 611 Part i: resolutions and derived functors 614 A3. 2 Free and Projective Modules 615 A3.3 Free and Projective Resolutions 617 A3.4 Injective Modules and Resolutions 618 A3. 4.1 Exercises 623 Injective Envelopes 623 Injective Modules over Noetherian Rings 623 A3.5 Basic Constructions with Complexes a3.5.1 Notation and definitions 626

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