Ind-abelian categories and quasi-coherent sheaves

TL;DR

This paper investigates conditions under which categories of ind-objects are abelian, generalizes weakly Tannakian categories, and applies these results to describe coherent sheaves on product schemes via Deligne tensor products.

Contribution

It provides new criteria for abelian categories of ind-objects and extends the concept of weakly Tannakian categories, with applications to coherent sheaves on fiber products of schemes.

Findings

Categories of ind-objects can be abelian under certain conditions.

The class of schemes with the resolution property is closed under fiber products.

Coherent sheaves on product schemes are described by Deligne tensor products.

Abstract

We study the question when a category of ind-objects is abelian. Our answer allows a further generalization of the notion of weakly Tannakian categories introduced by the author. As an application we show that, under suitable conditions, the category of coherent sheaves on the product of two schemes with the resolution property is given by the Deligne tensor product of the categories of coherent sheaves of the two factors. To do this we prove that the class of quasi-compact and semi-separated schemes with the resolution property is closed under fiber products.