Finite generation of exterior and symmetric powers

Abstract

If Q is a finitely generated abelian group, k a field, V a finitely generated kQ-module Bieri and Groves have proved that ^{[otimes ]m}V is a finitely generated as a kQ-module with diagonal Q-action if and only if ^∧iV is finitely generated as a kQ-module for all 1[les ]i[les ]m. We generalize this result by showing that if the mth exterior power of V or the mth symmetric power of V is finitely generated as a kQ-module so is the mth tensor power of V. Further we show the equivalence between the finite generation of symmetric and tensor powers in the case when the ground ring is a PID of characteristic 0.