Note on the linearisation of finite abelian groups

TL;DR

This paper investigates the conditions under which the group algebra of a finite abelian group over a commutative ring can be linearlyized, extending classical results using Gauss sums and exploring functorial properties.

Contribution

It proves the existence of a natural isomorphism between the group algebra and the dual group algebra for finite abelian groups over any commutative ring, generalizing previous results.

Findings

Established a natural isomorphism for finite abelian groups over any commutative ring.

Extended classical results beyond fields with roots of unity using Gauss sums.

Explored related functorial questions in the context of group algebra linearization.

Abstract

If $K$ is a field with enough roots of unity and $V$ an abelian group, the $K$-algebra $K[V]$ of the group $V$ is split semisimple, so that the canonical morphism $K[V]\to K^{V^\sharp}$, where $V^\sharp$ denotes the dual group of $V$ (which may be seen as Hom$(V,K^\times)$), is an isomorphism of $K$-algebras. If one removes the assumption that $K$ has enough roots of unity, one can easily deduce from it (by using a base change and Krull-Schmidt) that it remains a $K$-linear isomorphism $K[V]\xrightarrow{\simeq} K^{V^\sharp}$ natural in the group $V$ if one restricts to finite groups $V$ canceled by a fixed nonzero integer. The question of whether such an isomorphism, natural in the abelian group $V$, still exists without any other restriction than $V$ is finite and its order is invertible in $K$, is less obvious; we solve it positively, in a somewhat more general setting ($K$ being any commutative ring), by using Gauss sums. We also explore some related functorial questions.