Decompositions of homomorphisms of abelian $p$-groups with $p^2$-bounded target

Abstract

Let $p$ be a prime. We classify homomorphisms $H:V_1\to V_2$ between abelian $p$-groups that are direct sums of cyclic groups, under the assumption $p^2V_2=0$. We prove that every such homomorphism is isomorphic to a direct sum of finite indecomposable representations and that this decomposition is unique up to isomorphism and reordering of the summands.