# On flat generators and Matlis duality for quasicoherent sheaves

## TL;DR

This paper characterizes when the category of quasicoherent sheaves on a scheme has a flat generator, linking this property to the scheme being semiseparated and properties of injective objects.

## Contribution

It establishes the equivalence between the existence of a flat generator in QCoh(X), the exactness of internal hom functors into injectives, and the scheme being semiseparated.

## Key findings

- Flat generator exists iff scheme is semiseparated
- Internal hom functor exactness characterizes semiseparated schemes
- Provides new criteria for flat generators in quasicoherent sheaves

## Abstract

We show that for a quasicompact quasiseparated scheme $X$, the following assertions are equivalent: (1) the category $\operatorname{QCoh}(X)$ of all quasicoherent sheaves on $X$ has a flat generator; (2) for every injective object $\mathcal E$ of $\operatorname{QCoh}(X)$, the internal hom functor into $\mathcal E$ is exact; (3) the scheme $X$ is semiseparated.