Gorenstein homological algebra and universal coefficient theorems

We study criteria for a ring—or more generally, for a small category—to be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficient theorems found in the literature and to develop machinery for proving new ones. Among the universal coefficient theorems covered by our methods we find, besides all the classic examples, several exotic examples arising from the KK-theory of C*-algebras and also Neeman’s Brown–Adams representability theorem for compactly generated categories

Gorenstein homological algebra and universal coefficient theorems

We study criteria for a ring - or more generally, for a small category - to be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficient theorems found in the literature and to develop a machinery for proving new ones. Among the universal coefficient theorems covered by our methods we find, besides all the classic examples, several exotic examples arising from the KK-theory of C*-algebras and also Neeman's Brown-Adams representability theorem for compactly generated categories.Comment: 43 page

A Zariski-local notion of F-total acyclicity for complexes of sheaves

We study a notion of total acyclicity for complexes of flat sheaves over a scheme. It is Zariski-local - i.e. it can be verified on any open affine covering of the scheme - and it agrees, in their setting, with the notion studied by Murfet and Salarian for sheaves over a noetherian semi-separated scheme. As part of the study we recover, and in several cases extend the validity of, recent theorems on existence of covers and precovers in categories of sheaves. One consequence is the existence of an adjoint to the inclusion of these totally acyclic complexes into the homotopy category of complexes of flat sheaves

Categorical and homological aspects of module theory over commutative rings

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