Classifying exact categories via Wakamatsu tilting

TL;DR

This paper explores the classification of exact categories using Wakamatsu tilting, establishing embeddings into module categories and applications to artin algebra representation theory.

Contribution

It introduces a Morita-type embedding for exact categories with enough projectives/injectives and characterizes their images via Wakamatsu tilting and cotilting subcategories.

Findings

Exact categories with enough projectives can be embedded into module categories.

Wakamatsu tilting subcategories describe the images of these embeddings.

The ideal quotient of a module category by certain subcategories forms a torsionfree class.

Abstract

Using the Morita-type embedding, we show that any exact category with enough projectives has a realization as a (pre)resolving subcategory of a module category. When the exact category has enough injectives, the image of the embedding can be described in terms of Wakamatsu tilting (=semi-dualizing) subcategories. If moreover the exact category has higher kernels, then its image coincides with the category naturally associated with a cotilting subcategory up to summands. We apply these results to the representation theory of artin algebras. In particular, we show that the ideal quotient of a module category by a functorially finite subcategory closed under submodules is a torsionfree class of some module category.