The radical of functorially finite subcategories

TL;DR

This paper investigates the structure of functorially finite subcategories of module categories over artin algebras, characterizing when they have additive generators and finite representation type using radicals and morphism properties.

Contribution

It introduces the concept of the infinite radical for subcategories, providing new criteria for finite representation type and describing morphisms via irreducible morphisms.

Findings

A subcategory has an additive generator iff its infinite radical vanishes.

Finite representation type corresponds to noetherian and conoetherian conditions on morphism families.

Provides a nilpotency index of the radical independent of module length.

Abstract

Let $\Lambda$ be an artin algebra and $\mathcal{C}$ be a functorially finite subcategory of mod$\Lambda$ which contains $\Lambda$ or $D\Lambda$. We use the concept of the infinite radical of $\mathcal{C}$ and show that $\mathcal{C}$ has an additive generator if and only if rad$^\infty_{\mathcal{C}}$ vanishes. In this case we describe the morphisms in powers of the radical of $\mathcal{C}$ in terms of its irreducible morphisms. Moreover, under a mild assumption, we prove that $\mathcal{C}$ is of finite representation type if and only if any family of monomorphisms (epimorphisms) between indecomposable objects in $\mathcal{C}$ is noetherian (conoetherian). Also, by using injective envelopes, projective covers, left $\mathcal{C}$-approximations and right $\mathcal{C}$-approximations of simple $\Lambda$-modules, we give other criteria to describe whether $\mathcal{C}$ is of finite representation type. In addition, we give a nilpotency index of the radical of $\mathcal{C}$ which is independent from the maximal length of indecomposable $\Lambda$-modules in $\mathcal{C}$.