1st Edition

Covers and Envelopes in the Category of Complexes of Modules

By J.R. Garcia Rozas Copyright 1999
    152 Pages
    by Chapman & Hall

    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.

    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
    Introduction
    Spaltenstein's Quasi-Isomorphisms
    Exact, DG-Injective and DG Projective Covers and Envelopes
    Minimal Injective Resolutions
    Gorenstein Injective and Gorenstein Projective Complexes
    Preliminaries
    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
    Preliminaries
    Flat Pre-Envelope of Complexes
    Pure Injective and Cotorsion Envelopes
    Gorenstein Flat Complexes
    A Theorem on Perfect Rings
    DG-Pure Sequences
    Bibliography
    Index

    Biography

    J.R. Garcia Rozas (University of Almeria, Spain)