Torsion Freeness of Schur Modules

TL;DR

This paper extends conditions for torsion freeness from exterior and symmetric powers to a broad class of Schur modules over Noetherian rings, analyzing their structure and acyclicity.

Contribution

It generalizes torsion freeness criteria to Schur modules and studies the structure and acyclicity of Schur complexes for module maps.

Findings

Provided necessary and sufficient conditions for torsion freeness of Schur modules.

Explicitly computed radicals of ideals of maximal minors for Schur complexes.

Extended previous results from exterior and symmetric powers to a broader class of Schur modules.

Abstract

Let $R$ be a Noetherian commutative ring and $M$ an $R$-module with $\operatorname{pd_R} M\le 1$ that has rank. Necessary and sufficient conditions were provided by Lebelt for an exterior power $\wedge^k M$ to be torsion free. When $M$ is an ideal of $R$ similar necessary and sufficient conditions were provided by Tchernev for a symmetric power $S_k M$ to be torsion free. We extend these results to a broad class of Schur modules $L_{\lambda/\mu} M$. En route, for any map of finite free $R$ modules $\phi\: F\rightarrow G$ we also study the general structure of the Schur complexes $L_{\lambda/\mu}\phi$, and provide necessary and sufficient conditions for the acyclicity of any given $L_{\lambda/\mu}\phi$ by computing explicitly the radicals of the ideals of maximal minors of all its differentials.