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 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. IntroductionLet X be a scheme and Qco(X) the category of all quasi-coherent sheaves on X. A convenient way of approaching the derived category D(Qco(X)) goes back to Quillen [27], and consists in introducing a model category structure on C(Qco(X)), the category of unbounded chain complexes on Qco(X). In particular, one can compute morphisms between two objects X and Y of D(Qco(X)) as the C(Qco(X))morphisms between cofibrant and fibrant replacements of X and Y , respectively, modulo chain homotopy.Recently, Hovey has shown that model category structures naturally arise from small cotorsion pairs over C(Qco(X)), [20]. Since Qco(X) is a Grothendieck category [8], there is a canonical injective model category structure on C(Qco(X)). However, this structure is not monoidal, that is, compatible with the tensor product on Qco(X), [21, pp. 111-2]. Another natural, but not monoidal, model structure on C(Qco(X)) was constructed in [22] under the assumption of X being a Noetherian separated scheme with enough locally frees.The lack of compatibility with the tensor product was partially solved in [14, 25] by using flat quasi-coherent sheaves. The main result of [14] shows that in case X is quasi-compact and semi-separated, it is possible to construct a monoidal flat model structure on C(Qco(X)). The weak equivalences of this model structure are the same as the ones for the injective model structure, hence they induce the same cohomology functors (see [25] for a different approach). However, the structure of flat quasi-coherent sheaves is rather complex, and it is difficult to compute the associated fibrant and cofibrant replacements. Moreover, the methods of [14] depend heavily on the fact that the class of all flat modules is closed under direct limits.A different approach has recently been suggested in [10] for the particular case of quasi-coherent sheaves on the projective line P 1 (k). In that paper it was shown that the class of infinite-dimensional vector bundles (i.e., those quasi-coherent sheaves whose sections in all open affine sets are projective) imposes a monoidal model category structure on C(Qco (P 1 (k)). The proofs and techniques in [10] are strongly based on the Grothendiec...

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