Equivariant vector bundles, their derived category and $K$-theory on affine schemes

TL;DR

This paper investigates the structure of equivariant vector bundles on affine schemes with group actions, establishing conditions under which these bundles are extended from the base ring and linking derived category equivalences to $K$-theory isomorphisms.

Contribution

It proves that equivariant vector bundles on affine toric schemes are extended from the base ring in certain cases and connects derived category equivalences to isomorphisms in equivariant $K$-theory.

Findings

Equivariant vector bundles are extended from the base ring in specific cases.

Derived category equivalences imply isomorphisms in equivariant $K$-theory.

Results apply to affine schemes with affine group scheme actions.

Abstract

Let $G$ be an affine group scheme over a noetherian commutative ring $R$. We show that every $G$-equivariant vector bundle on an affine toric scheme over $R$ with $G$-action is extended from $\Spec(R)$ for several cases of $R$ and $G$. We show that given two affine schemes with group scheme actions, an equivalence of the equivariant derived categories implies isomorphism of the equivariant $K$-theories as well as equivariant $K'$-theories.