Categorical and homological aspects of module theory over commutative rings

Abstract

The purpose of this work is to understand the structure of the subcategories of mod(R) and the derived category D^b(R) for a commutative Noetherian ring R. Special focus is given to categories involving duality. We use these results to study homological dimension, maximal Cohen-Macaulay modules, and the singularities of a ring. Specifically, we classify certain resolving subcategories using semidualizing modules and also explore the relationship between these resolving subcategories and homological dimension. We also investigate the connections between semidualizing modules and rational singularities. Furthermore, using the theory of semidualizing modules and relative homological algebra, we prove a result on the depth formula. In order to construct Gersten-like complexes for singular schemes, we give an equivalence of derived categories. We also use this equivalence to study the Witt groups of categories associated to semidualizing modules. Lastly, we study the geometry of cohomological supports, a tool for understanding the thick subcategories over complete intersection rings. In particular, we show that when the Tor modules vanish, the cohomological support of the tensor product of two modules is the geometric join of the cohomological support of the original modules.