A functorial formalism for quasi-coherent sheaves on a geometric stack

TL;DR

This paper establishes an equivalence between categories of quasi-coherent sheaves on geometric stacks and comodules over Hopf algebroids, providing a functorial framework that generalizes classical scheme theory.

Contribution

It introduces a functorial formalism linking quasi-coherent sheaves on geometric stacks with Hopf algebroid comodules, extending classical scheme results.

Findings

Categories of quasi-coherent sheaves are equivalent to comodules over Hopf algebroids.

The category of quasi-coherent sheaves on a geometric stack is a Grothendieck category.

Constructs an adjunction for morphisms of stacks compatible with classical schemes.

Abstract

A geometric stack is a quasi-compact and semi-separated algebraic stack. We prove that the quasi-coherent sheaves on the small flat topology, Cartesian presheaves on the underlying category, and comodules over a Hopf algebroid associated to a presentation of a geometric stack are equivalent categories. As a consequence, we show that the category of quasi-coherent sheaves on a geometric stack is a Grothendieck category. We associate, in a 2-functorial way, to a 1-morphism of geometric stacks $f$, an adjunction $(f^*, f_*)$ for the corresponding categories of quasi-coherent sheaves that agrees with the classical one defined for schemes. This construction is described both geometrically in terms of the small flat site and algebraically in terms of the Hopf algebroid.