Covers and Envelopes in the Category of Complexes of Modules
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.
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apply argument automorphism boundary bounded class of objects clear commutative diagram consider construct Corollary cotorsion define Definition denote DG-cotorsion DG-flat DG-projective direct direct limits direct summand easy epimorphism equivalent exact complex exact cover example exists extension factored finite injective dimension finitely presented flat complex flat cover flat precover functor give Given going Gorenstein flat Gorenstein injective Gorenstein injective envelope Gorenstein projective cover Hence hereditary torsion theory Hom(C Hom(D homology homotopic hypothesis inductive injective complex injective resolution isomorphism L-cover L-precover left R-modules Lemma map of complexes minimal DG-injective complex module morphism n-Gorenstein ring natural object perfect preenvelope projective complex Proof properties Proposition prove pure injective quasi-isomorphism R-Mod Remark resolving result ring separable short exact sequence splits subcomplex Suppose surjective Theorem University verifies
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