Covers and Envelopes in the Category of Complexes of ModulesOver 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. |
Contents
Orthogonality in the Category of Complexes | 32 |
Gorenstein Injective and Projective Complexes | 52 |
52 | 78 |
Pure Sequences of Complexes | 108 |
| 133 | |
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Common terms and phrases
abelian category automorphism Card(S category of complexes class of objects closed under direct commutative diagram commutative noetherian ring complex F complex of left conclusion of Lemma Corollary cotorsion denote DG-cotorsion DG-injective envelope DG-pure direct limits direct sum direct summand E₁ epimorphism exact complex exact cover Ext¹(L finite injective dimension flat cover flat modules flat precover flat preenvelope Gorenstein flat complexes Gorenstein injective envelope Gorenstein projective complex Gorenstein projective cover Hence hereditary torsion theory Hom(C Hom(D Hom(E Hom(P homology isomorphism homotopic hypothesis I₁ injective cover injective module injective resolution isomorphism kernel L-cover left R-modules Lemma for Categories Let F M(id map of complexes minimal DG-injective complex monomorphism morphism n-Gorenstein ring noetherian ring projective module projective preenvelope Proof Proposition prove pure injective quasi-isomorphism R-Mod right coherent satisfies the conclusion separable functor sequence exact sequence of complexes short exact sequence subcomplex Suppose surjective Theorem Zorn's Lemma
References to this book
Abelian Groups, Rings, Modules, and Homological Algebra Pat Goeters,Overtoun M.G. Jenda Limited preview - 2006 |
Approximations and Endomorphism Algebras of Modules Rüdiger Göbel,Jan Trlifaj No preview available - 2006 |