Complexes and Diffrerential Graded Modules

Abstract: The main topic of the thesis is the generalization of some traditional module-theoretic homological applications to complexes of modules and to differential graded modules over differential graded rings. We introduce three possible generalizations of the classical notion of annihilator of an R-module. For linear functors D (R) -> D (R), preserving the triangulation, certain inclusion results for these annihilators are obtained. We study ascent and descent of Gorenstein and Cohen-Macaulay properties along a local homomorphism f:R -> S in the presence of a finite S-module which is of finite flat dimension over R thus generalizing the concept of homomorhism of finite flat dimension introduced by Luchezar Avramov and Hans-Bjørn Foxby. The two approaches of the classical homological algebra to homological dimensions (the resolutional approach and the functorial one) give rise to different invariants in the category of differential graded modules over a differential graded ring. We study this dichotomy and establish the simultaneous finiteness of the resolutional and the functorial flat dimension in one special case. We also generalize the notion of the canonical module of a Cohen-Macaulay ring to the case of a genuine differential graded ring.