Direct products of modules and the pure semisimplicity conjecture

TL;DR

This paper demonstrates that for certain rings, infinite direct products of non-isomorphic finitely generated modules cannot be decomposed into direct sums of finitely generated modules, impacting the pure semisimplicity conjecture.

Contribution

It establishes new resistance results for direct product decompositions over Artin algebras and certain noetherian domains, linking to the pure semisimplicity problem.

Findings

Infinite products of non-isomorphic finitely generated modules are not direct sums of finitely generated modules.

Results apply to Artin algebras and commutative noetherian domains of Krull dimension 1.

Implications for the pure semisimplicity conjecture are discussed.

Abstract

It is shown that, if $R$ is either an Artin algebra or a commutative noetherian domain of Krull dimension $1$, then infinite direct products of $R$-modules resist direct sum decomposition as follows: If $(M_n)_{n \in \Bbb N}$ is a family of non-isomorphic, finitely generated, indecomposable $R$-modules, then $\prod_{n\in \Bbb N} M_n$ is not a direct sum of finitely generated modules. The bearing of this direct product condition on the pure semisimplicity problem is discussed.