Homological dimensions for modules and complexes
Abstract
The projective dimension of Cartan and Eilenberg and the Gorenstein dimension of Auslander and Bridger are two classical homological dimensions for the class of finite modules over commutative noetherian local rings. Recently, new homological dimensions have been defined: complete intersection dimension by Avramov, Gasharov and Peeva, Cohen-Macaulay dimension and lower complete intersection dimension by Gerko. This thesis focuses on Gorenstein dimensions for modules and for complexes. In Chapter 1, we introduce and study a new homological dimension called upper Gorenstein dimension. It is modeled on the complete intersection dimension and has parallel properties. We show that this new dimension sits between Gorenstein dimension and complete intersection dimension and if one of these dimensions is finite, then it is equal to those to its left. We also construct families of modules for which strict inequality is obtained between various homological dimensions. By finding a module for which lower complete intersection dimension is strictly less than the complete intersection dimension, we answer a question of Gerko. In Chapter 2, we introduce and study a notion of Gorenstein projective dimension for arbitrary complexes of left modules over an associative ring. It specializes to Gorenstein projective dimension of Enochs and Jenda for left modules over an associative ring and to the Gorenstein projective dimension of Christensen for complexes with bounded below homology. Gorenstein projective dimension for complexes is less or equal to the projective dimension for complexes, notion defined by Avramov and Foxby. When the latter dimension is finite, equality is attained. These two dimensions have parallel properties. We also introduce and study a Tate cohomology theory for complexes of finite Gorenstein projective dimension. It specializes to existing cohomological theories, due to Tate, Farrell, Buchweitz, Avramov, Martsinkovsky and others. We show that Tate cohomology for complexes has most formal properties of earlier cohomologies of modules, in particular it admits a natural transformation to classical Ext groups. In the case of module arguments, we show that these maps fit into a long exact sequence, where every third term is a relative cohomology group defined for left modules of finite Gorenstein projective dimension.
Degree
Ph.D.
Advisors
Avramov, Purdue University.
Subject Area
Mathematics
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