Locally finitely presented categories with no flat objects

TL;DR

This paper investigates the relationship between different notions of flatness in locally finitely presented categories, showing that some categories, like quasi-coherent sheaves on projective space, have no nontrivial flat objects.

Contribution

It demonstrates that certain locally finitely presented categories lack non-zero flat objects, highlighting a divergence between categorical and stalk-based flatness notions.

Findings

Categories like Qcoh(P^n(R)) have only the zero object as a flat object.

There exist locally finitely presented categories with no nontrivial flat objects.

The paper clarifies the relationship between different flatness concepts in algebraic geometry.

Abstract

If $X$ is a quasi-compact and quasi-separated scheme, the category $Qcoh(X)$ of quasi-coherent sheaves on $X$ is locally finitely presented. Therefore categorical flat quasi-coherent sheaves naturally arise. But there is also the standard definition of flatness in $Qcoh(X)$ from the stalks. So it makes sense to wonder the relationship (if any) between these two notions. In this paper we show that there are plenty of locally finitely presented categories having no other categorical flats than the zero object. As particular instance, we show that $Qcoh(\mathbf{P}^n(R)))$ has no other categorical flat objects than zero, where $R$ is any commutative ring.