Submodule Categories of Wild Representation Type

Abstract

Let $Λ$ be a commutative local uniserial ring of length at least seven with radical factor ring $k$. We consider the category $S(Λ)$ of all possible embeddings of submodules of finitely generated $Λ$-modules and show that $S(Λ)$ is controlled $k$-wild with a single control object $I\in S(Λ)$. In particular, it follows that each finite dimensional $k$-algebra can be realized as a quotient $\End(X)/\End(X)_I$ of the endomorphism ring of some object $X\in S(Λ)$ modulo the ideal $\End(X)_I$ of all maps which factor through a finite direct sum of copies of $I$.