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
Zorn's Lemma for Categories
Torsion Theory Relative to Ext
Preserving and Reflecting Covers by Functors
Orthogonality in the Category of Complexes
Exact, DG-Injective and DG Projective Covers and Envelopes
Minimal Injective Resolutions
Gorenstein Injective and Gorenstein Projective Complexes
Gorenstein Injective Complexes
Gorenstein Projective Complexes
Flat and DG-Flat Complexes
First Definitions and Results
Some Canonical Isomorphisms
Flat Covers of Complexes
Existence of Flat Covers of Complexes over a Commutative Noetherian Ring with Finite Krull Dimension
Pure Sequences of Complexes
Flat Pre-Envelope of Complexes
Pure Injective and Cotorsion Envelopes
Gorenstein Flat Complexes
A Theorem on Perfect Rings