Equivariant Picard groups and Laurent polynomials

TL;DR

This paper investigates the structure of equivariant Picard groups of finite type algebras over fields, revealing their contraction properties and providing a natural decomposition for Laurent polynomial extensions.

Contribution

It establishes that equivariant Picard groups are contracted with a specific contraction related to étale cohomology, leading to a decomposition for Laurent polynomial rings.

Findings

Equivariant Picard groups are contracted with a specific contraction.

Provides a natural decomposition of Picard groups for Laurent polynomial extensions.

Connects algebraic K-theory with étale cohomology in the context of group actions.

Abstract

Let $G$ be a finite group. For a $G$-ring $A,$ let ${\rm Pic}^{\it G}({\it A})$ denote the equivariant Picard group of $A.$ We show that if $A$ is a finite type algebra over a field $k$ then ${\rm Pic}^{\it G}({\it A})$ is contracted in the sense of Bass with contraction $H_{et}^{1}(G; Spec(A), \mathbb{Z}).$ This gives a natural decomposition of the group ${\rm Pic}^{\it G}({\it A[t, t^{-1}]}).$