Duality and contravariant functors in the representation theory of artin algebras

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.