Exact subcategories, subfunctors of $\operatorname{Ext}$, and some applications

TL;DR

This paper develops methods to identify subfunctors of the Ext functor in exact categories, with applications in detecting regularity and studying Ulrich modules over local rings.

Contribution

It introduces new techniques for characterizing sub(bi)functors of Ext via additivity and restriction, and explores their applications in commutative algebra.

Findings

Characterization of subfunctors of Ext using additivity.

Identification of Ext subfunctors over local rings.

Applications to regularity detection and Ulrich modules.

Abstract

Let $(\mathcal{A},\mathcal{E})$ be an exact category. We establish basic results that allow one to identify sub(bi)functors of $\operatorname{Ext}_{\mathcal{E}}(-,-)$ using additivity of numerical functions and restriction to subcategories. We also study a small number of these new functors over commutative local rings in details, and find a range of applications from detecting regularity to understanding Ulrich modules.