Preface ......................................................... v
1 Dimension of a Local Ring .................................... 1
1.1 Nakayama's lemma ........................................ 1
1.2 Prime ideals ............................................ 2
1.3 Noetherian modules ...................................... 4
1.4 Modules of finite length ................................ 6
1.5 Hubert's basis theorem .................................. 8
1.6 Graded rings ............................................ 9
1.7 Filtered rings ......................................... 11
1.8 Local rings ............................................ 13
1.9 Regular local rings .................................... 17
2 Modules over a Local Ring ................................... 19
2.1 Support of a module .................................... 19
2.2 Associated prime ideals ................................ 20
2.3 Dimension of a module .................................. 22
2.4 Depth of a module ...................................... 24
2.5 Cohen-Macaulay modules ................................. 25
2.6 Modules of finite projective dimension ................. 27
2.7 The Koszul complex ..................................... 29
2.8 Regular local rings .................................... 31
2.9 Projective dimension and depth ......................... 32
2.10 J-depth ................................................ 34
2.11 The acyclicity theorem ................................. 36
2.12 An example ............................................. 39
3 Divisor Theory .............................................. 43
3.1 Discrete valuation rings ............................... 43
3.2 Normal domains ......................................... 44
3.3 Divisors ............................................... 46
3.4 Unique factorization ................................... 47
3.5 Torsion modules ........................................ 48
3.6 The first Chern class .................................. 49
3.7 Regular local rings .................................... 51
3.8 Picard groups .......................................... 51
3.9 Dedekind domains ....................................... 54
4 Completion .................................................. 57
4.1 Exactness of the completion functor .................... 57
4.2 Separation of the J-adic topology ...................... 59
4.3 Complete filtered rings ................................ 60
4.4 Completion of local rings .............................. 61
4.5 Structure of complete local rings ...................... 63
5 Injective Modules ........................................... 65
5.1 Injective modules ...................................... 65
5.2 Injective envelopes .................................... 67
5.3 Decomposition of injective modules ..................... 68
5.4 Matlis duality ......................................... 70
5.5 Minimal injective resolutions .......................... 73
5.6 Modules of finite injective dimension .................. 74
5.7 Gorenstein rings ....................................... 77
6 Local Cohomology ............................................ 81
6.1 Basic properties ....................................... 81
6.2 Local cohomology and dimension ......................... 84
6.3 Local cohomology and depth ............................. 84
6.4 Support in the maximal ideal ........................... 85
6.5 Local duality for Gorenstein rings ..................... 87
7 Dualizing Complexes ......................................... 89
7.1 Complexes of injective modules ......................... 89
7.2 Complexes with finitely generated cohomology ........... 93
7.3 The evaluation map ..................................... 96
7.4 Existence of dualizing complexes ....................... 98
7.5 The codimension function .............................. 100
7.6 Complexes of flat modules ............................. 102
7.7 Generalized evaluation maps ........................... 105
7.8 Uniqueness of dualizing complexes ..................... 107
8 Local Duahty ............................................... 109
8.1 Poincare series ....................................... 109
8.2 Grothendieck's local duality theorem .................. 113
8.3 Duality for Cohen-Macaulay modules .................... 117
8.4 Dualizing modules ..................................... 119
8.5 Locally factorial domains ............................. 121
8.6 Conductors ............................................ 122
8.7 Formal fibers ......................................... 125
9 Amplitude and Dimension .................................... 129
9.1 Depth of a complex .................................... 130
9.2 The dual of a module .................................. 136
9.3 The amphtude formula .................................. 137
9.4 Dimension of a complex ................................ 139
9.5 The tensor product formula ............................ 142
9.6 Depth inequalities .................................... 144
9.7 Condition Sr of Serre ................................. 148
9.8 Factorial rings and condition Sr ...................... 152
9.9 Condition S'r ......................................... 155
9.10 Specialization of Poincare series ..................... 158
10 Intersection Multiplicities ................................ 161
10.1 Introduction to Serre's conjectures ................... 161
10.2 Filtration of the Koszul complex ...................... 163
10.3 Euler characteristic of the Koszul complex ............ 167
10.4 A projection formula .................................. 170
10.5 Power series over a field ............................. 171
10.6 Power series over a discrete valuation ring ........... 175
10.7 Application of Cohen's structure theorem .............. 178
10.8 The amplitude inequality .............................. 181
10.9 Translation invariant operators ....................... 182
10.10 Todd operators ....................................... 184
10.11 Serre's conjecture in the graded case ................ 187
11 Complexes of Free Modules .................................. 189
11.1 McCoy's theorem ....................................... 189
11.2 The rank of a linear map .............................. 191
11.3 The Eisenbud-Buchsbaum criterion ...................... 194
11.4 Fitting's ideals ...................................... 196
11.5 The Euler characteristic .............................. 199
11.6 McRae's invariant ..................................... 203
11.7 The integral character of McRae's invariant ........... 205
Bibliography .................................................. 207
Index ......................................................... 211
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