Model category structures on chain complexes of sheaves (Q2706619)

Let \({\mathcal C}h(\mathcal{A})\) be the category of unbounded chain complexes in an abelian category \(\mathcal{A}\) obtained by adjoining formal inverses to all maps inducing homology isomorphisms. The category \({\mathcal C}h(\mathcal{A})\) is the unbounded derived category of \(\mathcal{A}\). A model structure on \({\mathcal C}h(\mathcal{A})\) is defined in the case where \(\mathcal{A}\) has a set of generators of finite projective dimension. The author uses this model structure to show, for instance: NEWLINENEWLINENEWLINE1. Suppose that \(S\) is a noetherian scheme with enough injectives and is either finite-dimensional or separated. Let \({\mathcal {QC}}o(S)\) be the category of quasi-coherent sheaves on \(S\). Then, locally free sheaves of finite rank form a set of small weak generators for the derived category \(D({\mathcal {QC}}o(S))\). NEWLINENEWLINENEWLINE2. Suppose that \((S,\mathcal{O})\) is a ringed space with finite hereditary global dimension. Then, NEWLINENEWLINENEWLINE(a) there is a cofibrantly generated proper model structure on \({\mathcal C}h(\mathcal{O}\text{-}{\mathcal M}od)\), called the flat model structure where weak equivalences are quasi-isomorphisms and the fibrations are the surjections with dimensionwise flasque kernel. NEWLINENEWLINENEWLINE(b) The flat model structure on \({\mathcal C}h(\mathcal{O}\text{-}{\mathcal M}od)\) makes it into a symmetric monoidal category.