Torsion in tensor powers of modules

TL;DR

This paper investigates torsion phenomena in tensor powers of modules over local rings, highlighting special modules like Frobenius powers that exhibit unique torsion properties.

Contribution

It introduces new insights into torsion behavior in tensor powers and characterizes modules with torsion in tensor products with arbitrary modules.

Findings

Tensor powers of modules often have nonzero torsion.

Frobenius powers of rings are key examples with specific torsion properties.

The study extends classical results on torsion in tensor products.

Abstract

Tensor products usually have nonzero torsion. This is a central theme of Auslander's paper "Modules over unramified regular local rings"; the theme continues in the work of Huneke and Wiegand. The main focus in this note is on tensor powers of a finitely generated module over a local ring. Also, we study torsion-free modules N with the property that its tensor product with any module M has torsion, unless M is very special. Important examples of such modules N are the Frobenius power of a ring, viewed as a module over itself, that is a complete intersection domain of positive characteristic.