Torsion Freeness 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.