Relative Homological Dimensions And Their Applications

In the middle of 1950s, Serre, Auslander and Buchsbaum proved that a commuta-tive Noetherian local ring is regular if and only if every finitely generated R-module has finite projective dimension. The Regularity Theorem opens a new chapter of investigat-ing commutative algebra via homological invariants. At the end of 1960s, Auslander found a new homological dimension-Gorenstein dimension available to the study of Gorenstein rings, which are very important in commutative algebra and algebraic ge-ometry, and related homological properties of module categories. Actually Gorenstein homological algebra has its root in this new dimension. Consequently, investigating the homological properties of module categories by different kinds of homological dimen-sions has become a very meaningful subject in commutative algebra and homological algebra.Inspired by this idea, this thesis aims at enriching some work already done in this area by some new homological dimensions. The thesis consists of four chapters.In Chapter 1, some main results and preliminaries are given.In Chapter 2, we provide five new criteria for a semidualizing module to be du-alizing, the first two results improve Takahashi et al.'s and Christensen's results re-spectively, another two results can be treated as generalizations of Jenda's and Xu's results, and the last theorem gives a new characterization of locally Gorenstein rings.Chapter 3 is devoted to the study of Cohen-Macaulay injective dimension and generalized local cohomology. We prove a vanishing theorem for generalized local cohomology. This result is applied to characterizations of regular and Gorenstein rings, and it is also useful to give a sufficient condition for a ring to be Cohen-Macaulay. This condition gives a partial answer to the following question of Takahashi [57]:Is a local ring Cohen-Macaulay if it admits a nonzero finitely generated module of finite Gorenstein injective dimension?In Chapter 4 we introduce the notions of strongly max-flat modules and related ho-mological dimensions over non-commutative rings. We show that the class of strongly max-flat R-modules and its Ext-orthogonal class form a perfect and hereditary cotor- sion pair. This result has some applications to characterizations of right SF, semisim-ple, left perfect, von Neumann regular and quasi-Frobenius rings.