Homological questions in local algebra / Jan R. Strooker.

Format:

Book

Author/Creator:

Strooker, Jan R., author.

Series:

London Mathematical Society lecture note series ; 145.

London Mathematical Society lecture note series ; 145

Language:

English

Subjects (All):

Commutative algebra.

Homology theory.

Physical Description:

1 online resource (xiii, 308 pages) : digital, PDF file(s).

Place of Publication:

Cambridge : Cambridge University Press, 1990.

Language Note:

English

Summary:

This book presents an account of several conjectures arising in commutative algebra from the pioneering work of Serre and Auslander-Buchsbaum. The approach is via Hochster's 'Big Cohen-Macaulay modules', though the complementary view point of Peskine-Szpiro and Roberts, who study the homology of certain complexes, is not neglected. Various refinements of Hochster's construction, obtained in collaboration with Bartijn, are included. A special feature is a long chapter written by Van den Dries which explains how a certain type of result can be 'lifted' from prime characteristic to characteristic zero. Though this is primarily a research monograph, it does provide introductions to most of the topics treated. Non-experts may therefore find it an appealing guide into an active area of algebra.

Contents:

Cover; Title; Copyright; Contents; Preface; Chapter 1 : HOMOLOGICAL PRELIMINARIES; 1.1 Acyclicity Lemma's; 1.2 A few isomorphisms of complexes; Chapter 2 : ADIC TOPOLOGIES AND COMPLETIONS; 2.1 Induced topologies and purity; 2.2 Completions; 2.3 Lifting in complete local rings; Chapter 3 : INJECTIVE ENVELOPES AND MINIMAL INJECTIVE RESOLUTIONS; 3.1 The injective envelope of a module; 3.2 Decomposition of injective modules; 3.3 Minimal injective resolutions; 3.4 Matlis Duality; Chapter 4 : LOCAL COHOMOLOGY AND KOSZUL COMPLEXES; 4.1 Local cohomology; 4.2 Koszul complexes

4.3 Limits of Koszul complexes and local cohomologyChapter 5 : (PRE-) REGULAR SEQUENCES AND DEPTH; 5.1 (Pre-) regular sequences; 5.2 Pre-regularity under completion; connections with local cohomology; 5.3 Depth; Chapter 6 : EXACTNESS OF COMPLEXES AND LINEAR EQUATIONS OVER RINGS; 6.1 The Acyclicity Lemma and a few consequences; 6.2 The Buchsbaum-Eisenbud criterion; 6.3 Linear equations over rings; Chapter 7 : COMPARING HOMOLOGICAL INVARIANTS; 7.1 Auslander-Buchsbaum - and Bass identities generalized; 7.2 Equalities and inequalities involving grade; 7.3 Annihilators of local cohomology

Chapter 8 : DIMENSION8.1 Krull's Hauptidealsatz; 8.2 Parameters and dimension; 8.3 Parameters and regular sequences; 8.4 Extensions of the Hauptidealsatz; 8.5 Dimension conjectures; Chapter 9 : COHEN-MACAULAY MODULES AND REGULAR RINGS; 9.1 Cohen-Macaulay modules and rings; 9.2 Regular local rings; 9.3 Complete local rings and the Direct Summand Conjecture; Chapter 10 : GORENSTEIN RINGS, LOCAL DUALITY, AND THE DIRECT SUMMAND CONJECTURE; 10.1 Gorenstein rings; 10.2 Local duality; 10.3 The Direct Summand and Monomial Conjectures in equal characteristic

Chapter 11 : FROBENIUS AND BIG COHEN-MACAULAY MODULES IN CHARACTERISTIC11.1 Modifications; 11.2 Relations between annihilators; 11.3 The Frobenius functor; 11.4 Pre-regular modules in characteristic p; 11.5 Balanced Big Cohen-Macaulay modules in characteristic; Chapter 12 : BIG COHEN-MACAULAY MODULES IN EQUAL CHARACTERISTIC; 12.0 Introduction; 12.1 Hochster algebras and equational constraints; 12.2 Constructible properties and transfer to finite fields; 12.3 Proof of Lemma IV; 12.4 The Weierstrass Theorems; 12.5 Henselian rings and henselization; 12.6 An approximation theorem of M. Artin

12.7 From noetherian to finitely generated algebrasChapter 13 : USES OF BIG COHEN-MACAULAY MODULES; 13.1 New Intersection Theorems and a few consequences; 13.2 Nonvanishing of Bass numbers; REFERENCES

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references.

ISBN:

1-139-88419-0

1-107-36619-4

1-107-37092-2

1-107-36128-1

1-107-36907-X

1-299-40399-9

1-107-36373-X

0-511-62924-9