Quillen equivalent models for the derived category of flats and the resolution property

TL;DR

This paper establishes Quillen equivalences between different models of derived categories of flats, very flats, and infinite-dimensional vector bundles on certain schemes, under specific assumptions.

Contribution

It introduces conditions under which the derived category of flats is equivalent to that of very flats and vector bundles, extending the understanding of model categories in algebraic geometry.

Findings

Derived categories of flats and very flats are equivalent on certain schemes.

Schemes with the resolution property have derived categories equivalent to those of infinite-dimensional vector bundles.

The equivalences are established via Quillen model category theory.

Abstract

We investigate under which assumptions a subclass of flat quasi-coherent shea\-ves on a quasi-compact and semi-separated scheme allows to "mock" the homotopy category of projective modules. Our methods are based on module theoretic properties of the subclass of flat modules involved as well as their behaviour with respect to Zariski localizations. As a consequence we get that, for such schemes, the derived category of flats is equivalent to the derived category of very flats. If, in addition, the scheme satisfies the resolution property then both derived categories are equivalent to the derived category of infinite-dimensional vector bundles. The equivalences are inferred from a Quillen equivalence between the corresponding models.