Torsion Freeness of Schur Modules

2018/08/02 by Allahverdi, Muberra, Tchernev, Alexandre
#Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1808.00602

Abstract

Let R be a Noetherian commutative ring and M an R-module with pd_R M≤ 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_λ/μ M. En route, for any map of finite free R modules ϕ F→ G we also study the general structure of the Schur complexes L_λ/μϕ, and provide necessary and sufficient conditions for the acyclicity of any given L_λ/μϕ by computing explicitly the radicals of the ideals of maximal minors of all its differentials.

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