Covers and Envelopes in the Category of Complexes of Modules

by Garcia Rozas; J.R.

• ISBN13: 9781584880042
• ISBN10: 158488004X
• eBook ISBN(s): 9781351457606
• Format: Nonspecific Binding
• Copyright: 1999-05-11
• Publisher: Chapman & Hall/

Summary

Over the last few years, the study of complexes has become increasingly important. To date, however, most of the research is scattered throughout the literature or available only as lecture notes. Covers and Envelopes in the Category of Complexes of Modules collects these scattered notes and results into a single, concise volume that provides an account of recent developments in the theory and presents several new and important ideas.The author introduces the theory of complexes of modules using only elementary tools-making the field more accessible to non-specialists. He focuses the study on envelopes and covers in this category with respect to some well established and important classes of complexes. He places particular emphasis on DG-injective and DG-projective complexes and flat and DG-flat covers.Other topics covered include Zorn's Lemma for categories, preserving and reflecting covers by functors, orthogonality in the category of complexes, Gorenstein injective and projective complexes, and pure sequences of complexes.Along with its value as a collection of recent work in the field, Covers and Envelopes in the Category of Complexes of Modules presents powerful new ideas that will undoubtedly advance homological methods. Mathematicians-especially researchers in module theory and homological algebra-will welcome this volume as a reference guide and for its new and important results.

Table of Contents

Covers and Envelopes in Categories	1	(31)
Zorn's Lemma for Categories	1	(5)
Torsion Theory Relative to Ext	6	(15)
The Definition of Torsion Theory Relative to Ext	6	(5)
Examples	11	(1)
Hereditary and Perfect Torsion Theories	12	(7)
Torsion Theories and Balance	19	(2)
Preserving and Reflecting Covers by Functors	21	(11)
Orthogonality in the Category of Complexes	32	(20)
Introduction	32	(2)
Spaltenstein's Quasi-Isomorphisms	34	(4)
Exact, DG-Injective and DG-Projective Covers and Envelopes	38	(9)
Minimal Injective Resolutions	47	(5)
Gorenstein Injective and Projective Complexes	52	(29)
Preliminaries	52	(3)
Gorenstein Injective Complexes	55	(14)
Existence of Envelopes	60	(4)
Existence of DG-Injective and Gorenstein Injective Covers	64	(5)
Gorenstein Projective Complexes	69	(12)
Auslander's Last Theorem	74	(7)
Flat and DG-Flat Complexes	81	(27)
First Definitions and Results	81	(6)
Some Canonical Isomorphisms	87	(9)
Flat Covers of Complexes	96	(4)
Existence of Flat Covers of Complexes Over ZZ	97	(1)
Cotorsion Complexes	98	(2)
Existence of Flat Covers of Complexes Over a Commutative Noetherian Ring with Finite Krull Dimension	100	(8)
DG-Flat Covers	104	(4)
Pure Sequences of Complexes	108	(25)
Preliminaries	108	(2)
Flat Preenvelope of Complexes	110	(3)
Pure Injective and Cotorsion Envelopes	113	(4)
Gorenstein Flat Complexes	117	(7)
Existence of Gorenstein Flat Covers	122	(2)
A Theorem on Perfect Rings	124	(3)
DG-Pure Sequences	127	(6)
Bibliography	133	(2)
Index	135