## Computational Methods in Commutative Algebra and Algebraic GeometryFrom the reviews: "... Many parts of the book can be read by anyone with a basic abstract algebra course... it was one of the author's intentions to equip students who are interested in computational problems with the necessary algebraic background in pure mathematics and to encourage them to do further research in commutative algebra and algebraic geometry. But researchers will also benefit from this exposition. They will find an up-to-date description of the related research ... The reviewer recommends the book to anybody who is interested in commutative algebra and algebraic geometry and its computational aspects." Math. Reviews 2002 "... a sophisticated notebook, with plenty of suggestions, examples and cross references ... It is a welcome new and deep exploration into commutative algebra and its relations with algebraic geometry. It is full of results, from simple tricks to more elaborate constructions, all having in common a computational and constructive nature..." Jahresberichte der DMV 1999 |

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### Contents

Fundamental Algorithms | 7 |

11 Grobner Basics | 8 |

12 Division Algorithms | 12 |

13 Computation of Syzygies | 18 |

14 Hilbert Functions | 21 |

15 Computer Algebra Systems | 26 |

Toolkit | 29 |

21 Elimination Techniques | 30 |

66 Integral Closure of an Ideal | 176 |

67 Integral Closure of a Morphism | 184 |

Ideal Transforms and Rings of Invariants | 189 |

71 Divisorial Properties of Ideal Transforms | 190 |

72 Equations of Blowup Algebras | 193 |

73 Subrings | 202 |

74 Rings of Invariants | 209 |

Computation of Cohomology | 219 |

22 Rings of Endomorphisms | 35 |

23 Noether Normalization | 37 |

24 Fitting Ideals | 41 |

25 Finite and QuasiFinite Morphisms | 46 |

26 Flat Morphisms | 49 |

27 CohenMacaulay Algebras | 58 |

Principles of Primary Decomposition | 65 |

31 Associated Primes and Irreducible Decomposition | 67 |

32 Equidimensional Decomposition of an Ideal | 77 |

33 Equidimensional Decomposition Without Exts | 83 |

34 Mixed Primary Decomposition | 85 |

35 Elements of Factorizers | 90 |

Computing in Artin Algebras | 103 |

41 Structure of Artin Algebras | 104 |

42 ZeroDimensional Ideals | 109 |

43 Idempotents versus Primary Decomposition | 113 |

44 Decomposition via Sampling | 115 |

45 Root Finders | 120 |

Nullstellensatze | 127 |

51 Radicals via Elimination | 128 |

52 Modules of Differentials and Jacobian Ideals | 130 |

53 Generic Socles | 134 |

54 Explicit Nullstellensatze | 136 |

55 Finding Regular Sequences | 141 |

56 Top Radical and Upper Jacobians | 146 |

Integral Closure | 149 |

61 Integrally Closed Rings | 151 |

62 Multiplication Rings | 154 |

63 S2ification of an Affine Ring | 159 |

64 Desingularization in Codimension One | 167 |

65 Discriminants and Multipliers | 173 |

81 Eyeballing | 220 |

82 Local Duality | 222 |

83 Approximation | 224 |

Degrees of Complexity of a Graded Module | 227 |

91 Degrees of Modules | 230 |

92 Index of Nilpotency | 244 |

93 Qualitative Aspects of Noether Normalization | 249 |

94 Homological Degrees of a Module | 263 |

95 Complexity Bounds in Local Rings | 273 |

Primer on Commutative Algebra | 281 |

A2 Krull Dimension | 288 |

A3 Graded Algebras | 295 |

A4 Integral Extensions | 298 |

A5 Finitely Generated Algebras over Fields | 305 |

A6 The Method of Syzygies | 309 |

A7 CohenMacaulay Rings and Modules | 321 |

A8 Local Cohomology | 329 |

A9 Linkage Theory | 338 |

Hilbert Functions | 343 |

B2 The Study of R via grFR | 347 |

B3 The HilbertSamuel Function | 352 |

B4 Hilbert Functions Resolutions and Local Cohomology | 356 |

B5 Lexsegment Ideals and Macaulay Theorem | 359 |

B6 The Theorems of Green and Gotzmann | 362 |

Using Macaulay 2 | 367 |

C1 Elementary Uses of Macaulay 2 | 368 |

C2 Local Cohomology of Graded Modules | 382 |

C3 Cohomology of a Coherent Sheaf Mathematical Background | 387 |

393 | |

405 | |