Approximation of sheaves on algebraic stacks

TL;DR

This paper extends the characterization of certain morphisms from affine schemes to non-affine cases and demonstrates that on algebraic stacks, every quasi-coherent sheaf can be expressed as a union of finitely generated subsheaves.

Contribution

It provides a new characterization for non-affine morphisms and proves a structural property of quasi-coherent sheaves on algebraic stacks.

Findings

Characterization of non-affine morphisms similar to affine case

Every quasi-coherent sheaf is a union of finitely generated subsheaves

Applicable to quasi-compact and quasi-separated algebraic stacks

Abstract

Raynaud--Gruson characterized flat and pure morphisms between affine schemes in terms of projective modules. We give a similar characterization for non-affine morphisms. As an application, we show that every quasi-coherent sheaf is the union of its finitely generated quasi-coherent subsheaves on any quasi-compact and quasi-separated algebraic stack.