Model category structures arising from Drinfeld vector bundles

By Sergio Estrada, P. Asensio, Mike Prest, Jan Trlifaj. 2009

48 citations recorded by the source corpus.

We present a general construction of model category structures on the category $\Ch(\Qco(X))$ 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 $\mathbb D (\Qco(X))$ via various model structures on $\Ch(\Qco(X))$. 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/37564af2c3d6f30dd83b6d3f4550aa8118345889

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves (2013, background): “The exact model categories give in many respects a complementary approach to that of dg algebras, with different advantages and weaknesses, and it has a good potential for instance in Gorenstein homological algebra [39], homological algebra in the category of quasi-coherent sheaves [22, 23, 28], or ”
Almost free modules and Mittag-Leffler conditions (2009, methodology): “In [ 15 ], Gillespie’s result was extended further, to deconstructible classes, for quasi– coherent sheaves on a semi–separated scheme X. However, as mentioned in the Introduction, deconstructibility is also a necessary condition for making Hovey’s approach from [25] applicable in this setting.”
Deconstructibility and the Hill Lemma in Grothendieck categories (2010, background): “If one wishes to deal with sheaves, good news coming from [13] is that one can prove deconstructibility of a class of sheaves locally: Theorem.”
On exact categories and applications to triangulated adjoints and model structures (2010, methodology): “This is closely related to the work of Gillespie [18, 19, 20], and Estrada, Guil, Prest and Trlifaj [16].”