Model category structures arising from Drinfeld vector bundles

Abstract

We present a general construction of model category structures on the category C(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 D(Qco(X)) via various model structures on C(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 restricted Drinfeld vector bundles. Finally, we prove that the unrestricted case does not induce a model category structure as above in general.