Tensor generators on schemes and stacks

TL;DR

This paper characterizes algebraic stacks with affine stabilizers that satisfy the resolution property, linking it to quotient structures and tensor generators, thus generalizing previous results to broader classes of schemes and stacks.

Contribution

It extends Totaro's result to non-normal and non-noetherian schemes and stacks, establishing equivalences between the resolution property, quotient structures, and tensor generators.

Findings

Resolution property characterized by quotient of quasi-affine schemes

Equivalence between vector bundles inducing quotient structures and tensor generators

Generalization of Totaro's result to broader classes of schemes and stacks

Abstract

We show that an algebraic stack with affine stabilizer groups satisfies the resolution property if and only if it is a quotient of a quasi-affine scheme by the action of the general linear group, or equivalently, if there exists a vector bundle whose associated frame bundle has quasi-affine total space. This generalizes a result of B. Totaro to non-normal and non-noetherian schemes and algebraic stacks. Also, we show that the vector bundle induces such a quotient structure if and only if it is a tensor generator in the category of quasi-coherent sheaves.