Invariant ideals of abelian group algebras under the torus action of a field, II

Abstract

Let V = V 1 ⊕ V 2 be a finite-dimensional vector space over an infinite locally-finite field F. Then V admits the torus action of G = F • by defining ( v 1 ⊕ v 2 ) g = v 1 g − 1 ⊕ v 2 g . If K is a field of characteristic different from that of F, then G acts on the group algebra K [ V ] and it is an interesting problem to determine all G-stable ideals of this algebra. In this paper, we show that, for almost all fields F, the G-stable ideals are uniquely writable as finite irredundant intersections of augmentation ideals of subspaces W 1 ⊕ W 2 , with W 1 ⊆ V 1 and W 2 ⊆ V 2 . As a consequence, the set of all G-stable ideals is Noetherian.