Gorenstein homological algebra and universal coefficient theorems

TL;DR

This paper develops a unified framework for Gorenstein homological algebra and universal coefficient theorems, covering classical and exotic cases including KK-theory and Brown-Adams representability.

Contribution

It introduces criteria for Gorenstein properties of rings and categories, and creates a machinery to prove new universal coefficient theorems beyond existing literature.

Findings

Unified criteria for Gorenstein rings and categories.

A general machinery for proving universal coefficient theorems.

Application to exotic examples from KK-theory and Brown-Adams theorem.

Abstract

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.