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E-book: Lecture Notes On Local Rings

(Aarhus Univ, Denmark), Edited by (Aarhus Univ, Denmark)
  • Format: 224 pages
  • Pub. Date: 20-Jun-2014
  • Publisher: World Scientific Publishing Co Pte Ltd
  • Language: eng
  • ISBN-13: 9789814603676
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Lecture Notes On Local RingsLarger Image
  • Format: 224 pages
  • Pub. Date: 20-Jun-2014
  • Publisher: World Scientific Publishing Co Pte Ltd
  • Language: eng
  • ISBN-13: 9789814603676
Other books in subject:

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"The content in Chapter 1-3 is a fairly standard one-semester course on local rings with the goal to reach the fact that a regular local ring is a unique factorization domain. The homological machinery is also supported by Cohen-Macaulay rings and depth.In Chapters 4-6 the methods of injective modules, Matlis duality and local cohomology are discussed. Chapters 7-9 are not so standard and introduce the reader to the generalizations of modules to complexes of modules. Some of Professor Iversen's results are given in Chapter 9. Chapter 10 is about Serre's intersection conjecture. The graded case is fully exposed. The last chapter introduces the reader to Fitting ideals and McRae invariants."--
Preface v
1 Dimension of a Local Ring
1(18)
1.1 Nakayama's lemma
1(1)
1.2 Prime ideals
2(2)
1.3 Noetherian modules
4(2)
1.4 Modules of finite length
6(2)
1.5 Hilbert's basis theorem
8(1)
1.6 Graded rings
9(2)
1.7 Filtered rings
11(2)
1.8 Local rings
13(4)
1.9 Regular local rings
17(2)
2 Modules over a Local Ring
19(24)
2.1 Support of a module
19(1)
2.2 Associated prime ideals
20(2)
2.3 Dimension of a module
22(2)
2.4 Depth of a module
24(1)
2.5 Cohen-Macaulay modules
25(2)
2.6 Modules of finite projective dimension
27(2)
2.7 The Koszul complex
29(2)
2.8 Regular local rings
31(1)
2.9 Projective dimension and depth
32(2)
2.10 D-depth
34(2)
2.11 The acyclicity theorem
36(3)
2.12 An example
39(4)
3 Divisor Theory
43(14)
3.1 Discrete valuation rings
43(1)
3.2 Normal domains
44(2)
3.3 Divisors
46(1)
3.4 Unique factorization
47(1)
3.5 Torsion modules
48(1)
3.6 The first Chern class
49(2)
3.7 Regular local rings
51(1)
3.8 Picard groups
51(3)
3.9 Dedekind domains
54(3)
4 Completion
57(8)
4.1 Exactness of the completion functor
57(2)
4.2 Separation of the J-adic topology
59(1)
4.3 Complete filtered rings
60(1)
4.4 Completion of local rings
61(2)
4.5 Structure of complete local rings
63(2)
5 Injective Modules
65(16)
5.1 Injective modules
65(2)
5.2 Injective envelopes
67(1)
5.3 Decomposition of injective modules
68(2)
5.4 Matlis duality
70(3)
5.5 Minimal injective resolutions
73(1)
5.6 Modules of finite injective dimension
74(3)
5.7 Gorenstein rings
77(4)
6 Local Cohomology
81(8)
6.1 Basic properties
81(3)
6.2 Local cohomology and dimension
84(1)
6.3 Local cohomology and depth
84(1)
6.4 Support in the maximal ideal
85(2)
6.5 Local duality for Gorenstein rings
87(2)
7 Dualizing Complexes
89(20)
7.1 Complexes of injective modules
89(4)
7.2 Complexes with finitely generated cohomology
93(3)
7.3 The evaluation map
96(2)
7.4 Existence of dualizing complexes
98(2)
7.5 The codimension function
100(2)
7.6 Complexes of flat modules
102(3)
7.7 Generalized evaluation maps
105(2)
7.8 Uniqueness of dualizing complexes
107(2)
8 Local Duality
109(20)
8.1 Poincare series
109(4)
8.2 Grothendieck's local duality theorem
113(4)
8.3 Duality for Cohen--Macaulay modules
117(2)
8.4 Dualizing modules
119(2)
8.5 Locally factorial domains
121(1)
8.6 Conductors
122(3)
8.7 Formal fibers
125(4)
9 Amplitude and Dimension
129(32)
9.1 Depth of a complex
130(6)
9.2 The dual of a module
136(1)
9.3 The amplitude formula
137(2)
9.4 Dimension of a complex
139(3)
9.5 The tensor product formula
142(2)
9.6 Depth inequalities
144(4)
9.7 Condition Sr of Serre
148(4)
9.8 Factorial rings and condition Sr
152(3)
9.9 Condition S'r
155(3)
9.10 Specialization of Poincare series
158(3)
10 Intersection Multiplicities
161(28)
10.1 Introduction to Serre's conjectures
161(2)
10.2 Filtration of the Koszul complex
163(4)
10.3 Euler characteristic of the Koszul complex
167(3)
10.4 A projection formula
170(1)
10.5 Power series over a field
171(4)
10.6 Power series over a discrete valuation ring
175(3)
10.7 Application of Cohen's structure theorem
178(3)
10.8 The amplitude inequality
181(1)
10.9 Translation invariant operators
182(2)
10.10 Todd operators
184(3)
10.11 Serre's conjecture in the graded case
187(2)
11 Complexes of Free Modules
189(18)
11.1 McCoy's theorem
189(2)
11.2 The rank of a linear map
191(3)
11.3 The Eisenbud-Buchsbaum criterion
194(2)
11.4 Fitting's ideals
196(3)
11.5 The Euler characteristic
199(4)
11.6 McRae's invariant
203(2)
11.7 The integral character of McRae's invariant
205(2)
Bibliography 207(4)
Index 211
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