Endomorphism algebras over commutative rings and torsion in self tensor products

TL;DR

This work investigates torsion phenomena in self-tensor products of finitely generated modules over a Noetherian local ring by leveraging the natural End_R(M)-action and the possibility of End_R(M) forming an $R^*$-algebra. It proves that for an indecomposable torsion-free module $M$ of finite rank, if $E=\End_R(M)$ carries an $R^*$-algebra structure and $M_* \otimes_E M$ is torsion-free, then $M$ must be a cyclic $E$-module, provided $R$ is a domain or, in characteristic 2, reduced; the key step is showing a torsion submodule generated by $x \otimes y - a\, y \otimes x$ with $a^2=1$ must vanish. The paper also gives an explicit example showing that $M$ and $M$ may be torsion-free while $M_* \otimes_E M$ remains torsion, illustrating that endomorphism-based obstructions can be strictly sharper than those arising from $M \otimes_R M$. Overall, the results extend Huneke–Wiegand-type phenomena from ideals to higher-rank modules and highlight when end-based torsion is the decisive obstruction.

Abstract

Let $R$ be a commutative Noetherian local ring. We study tensor products involving a finitely generated $R$-module $M$ through the natural action of its endomorphism ring. In particular, we study torsion properties of self tensor products in the case where $\operatorname{End}_R(M)$ has an $R^*$-algebra structure, and prove that if $M$ is indecomposable, then $M \otimes_{\operatorname{End}_R(M)} M$ must always have torsion in this case under mild hypotheses.

Theorems & Definitions (21)

• Theorem 1.1
• Proposition 2.1: HW94
• Definition 2.2
• Example 2.3
• Definition 2.4
• Proposition 2.5
• proof
• Remark 2.6
• Proposition 3.1
• proof