Model category structures arising from Drinfeld vector bundles

TL;DR

This paper develops a new method to construct model category structures on complexes of quasi-coherent sheaves over schemes, extending previous approaches and including cases like Drinfeld vector bundles, with applications to derived categories.

Contribution

It introduces a general construction of model structures on complexes of quasi-coherent sheaves that does not rely on closure under direct limits, encompassing Drinfeld vector bundles and related sheaves.

Findings

Constructed model structures on chain complexes of quasi-coherent sheaves.

Recovered recent flat model structures for quasi-coherent sheaves.

Showed the non-existence of certain model structures in the unrestricted case.

Abstract

We present a general construction of model category structures on the category $\mathbb{C}(\mathfrak{Qco}(X))$ of unbounded chain complexes of quasi-coherent sheaves on a semi-separated scheme $X$. The construction is based on making compatible the filtrations of individual modules of sections at open affine subsets of $X$. It does not require closure under direct limits as previous methods. We apply it to describe the derived category $\mathbb D (\mathfrak{Qco}(X))$ via various model structures on $\mathbb{C}(\mathgrak{Qco}(X))$. As particular instances, we recover recent results on the flat model structure for quasi-coherent sheaves. Our approach also includes the case of (infinite-dimensional) vector bundles, and of restricted flat Mittag-Leffler quasi-coherent sheaves, as introduced by Drinfeld. Finally, we prove that the unrestricted case does not induce a model category structure as above in general.