The flat model structure on complexes of sheaves

By James Gillespie. 2006

51 citations recorded by the source corpus.

Let Ch(O) be the category of chain complexes of 0-modules on a topological space T (where O is a sheaf of rings on T). We put a Quillen model structure on this category in which the cofibrant objects are built out of flat modules. More precisely, these are the dg-flat complexes. Dually, the fibrant objects will be called dg-cotorsion complexes. We show that this model structure is monoidal, solving the previous problem of not having any monoidal model structure on Ch(O). As a corollary, we have a general framework for doing homological algebra in the category Sh(0) of 0-modules. I.e., we have a natural way to define the functors Ext and Tor in Sh(O).

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f6430c70ceb6b279a6891f677c88eab17683a10a

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

In Statistical and Nonlinear Physics

Citations in the corpus

Covers in finitely accessible categories (2010, methodology): “We point out the work of Aldrich, Enochs, Garćıa Rozas and Oyonarte [2] and El Bashir [14] in arbitrary Grothendieck categories, Enochs, Estrada, Garćıa Rozas and Oyonarte for quasi-coherent sheaves over the projective line [16] and for quasi-coherent sheaves over a scheme [17], Enochs and Oyonarte ”
Deconstructibility and the Hill Lemma in Grothendieck categories (2010, background): “For example in: (1) Gillespie’s construction [16, 17, 18] of flat monoidal model structures on module and sheaf categories (see [23] for a nice overview).”
More Concise Algebraic Topology: Localization, Completion, and Model Categories (2012, background): “Moreover, there is a ‘flat” model structure that is monoidal in such contexts [65, 49, 50].”
On exact categories and applications to triangulated adjoints and model structures (2010, methodology): “The main incentive of Gillespie in his work [18, 19, 20] was to give a clean and general framework to deal with the derived functor of ⊗.”
Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves (2013, methodology): “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 ”
Enhanced six operations and base change theorem for higher Artin stacks (2012, unknown): “The flat model structure for a ringed topological space has been constructed by [14,15].”