Duality and contravariant functors in the representation theory of artin algebras

TL;DR

This paper explores the algebraic characterization of certain functors in the representation theory of artin algebras, linking model theory, definable categories, and duality, and extends results to broader categories.

Contribution

It provides algebraic characterizations of contravariant functors related to definable categories over artin algebras and generalizes these results to locally finitely presented categories.

Findings

Characterization of additive functors preserving inverse limits

Finitely presented functors correspond to exact sequences involving modules

Extension of results to categories with dualising varieties

Abstract

We know that the model theory of modules leads to a way of obtaining definable categories of modules over a ring $R$ as the kernels of certain functors $(R\textbf{-Mod})^{\text{op}}\to\textbf{Ab}$ rather than of functors $R\textbf{-Mod}\to\textbf{Ab}$ which are given by a pp pair. This paper will give various algebraic characterisations of these functors in the case that $R$ is an artin algebra. Suppose that $R$ is an artin algebra. An additive functor $G:(R\textbf{-Mod})^{\text{op}}\to\textbf{Ab}$ preserves inverse limits and $G|_{(R\textbf{-mod})^{\text{op}}}:(R\textbf{-mod})^{\text{op}}\to\textbf{Ab}$ is finitely presented if and only if there is a sequence of natural transformations $(-,A)\to(-,B)\to G\to 0$ for some $A,B\in R\textbf{-mod}$ which is exact when evaluated at any left $R$-module. Any additive functor $(R\textbf{-Mod})^{\text{op}}\to\textbf{Ab}$ with one of these equivalent properties has a definable kernel, and every definable subcategory of $R\textbf{-Mod}$ can be obtained as the kernel of a family of such functors. In the final section a generalised setting is introduced, so that our results apply to more categories than those of the form $R\textbf{-Mod}$ for an artin algebra $R$. That is, our results are extended to those locally finitely presented $K$-linear categories whose finitely presented objects form a dualising $K$-variety, where $K$ is a commutative artinian ring.