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The homotopy theory of dg-categories and derived Morita theory. (English) Zbl 1118.18010

Morita theory for rings states that any functor from \(A\)-modules to \(B\)-modules commuting with colimits is necessarily the tensor product with some \(A^{\text{op}} \otimes B\)-module. When \(A\) and \(B\) are dg-algebras this is not so anymore as illustrated by D. Dugger and B. Shipley [Duke Math. J. 124, 587–617 (2004; Zbl 1056.19002)]. To fix this problem, the author proposes to work instead with dg-categories, i.e., categories enriched over chain complexes (when the category has a single object this is precisely a dg-algebra). G. Tabuada proved [C. R. Math. Acad. Sci. Paris 340, 15–19 (2005; Zbl 1060.18010)] that dg-categories form a model category where the weak equivalences are the analogues to the Dwyer-Kan equivalences for simplicial categories, see J. E. Bergner [Trans. Am. Math. Soc. 359, 2043–2058 (2007; Zbl 1114.18006)].
In this article a handy model for mapping spaces is identified in terms of modules. Given a dg-category \({\mathcal C}\), a \({\mathcal C}\)-module consists of chain complexes \(F(x)\) for all objects \(x\) in \({\mathcal C}\) and morphisms of chain complexes \({\mathcal C}(x, y) \otimes F(x) \rightarrow F(y)\). The category of \({\mathcal C}\)-modules is again a model category where the weak equivalences are objectwise quasi-isomorphisms of chain complexes. Among the \({\mathcal C}\otimes {\mathcal D}^{\text{op}}\)-modules consider the representable ones, i.e., those for which \(F(x, -)\), for any \(x \in {\mathcal C}\), is of the form \({\mathcal D}(-, y)\) for some \(y \in {\mathcal D}\). Define then \(\mathcal F({\mathcal C}, {\mathcal D})\) to be the category which has as objects the modules weakly equivalent to a representable one and as morphisms the quasi-isomorphisms. Then the mapping space \(\text{Map}({\mathcal C}, {\mathcal D})\) is weakly equivalent to the nerve \(N(\mathcal F({\mathcal C}, {\mathcal D}))\).
The derived tensor product of dg-categories induces a symmetric monoidal structure on the homotopy category of dg-categories. It is shown to be closed monoidal, so there exist dg-categories \({\mathbb R} \operatorname{Hom}({\mathcal C}, {\mathcal D})\). To develop Morita theory in the context of dg-categories, the author considers the dg-category \(\text{Int}({\mathcal C}(k))\) of cofibrant chain complexes and identifies the dg-category of cofibrant \({\mathcal C}^{\text{op}}\)-modules with \(\widehat{\mathcal C} = \mathbb R \operatorname{Hom}(C^{\text{op}}, {\mathcal I}nt({\mathcal C}(k)))\). He shows then that the full sub-dg-category \(\mathbb R \operatorname{Hom}_c (\widehat{\mathcal C}, \widehat{\mathcal D})\) of \(\mathbb R \operatorname{Hom}(\widehat{\mathcal C}, \widehat{\mathcal D})\) of morphisms commuting with infinite direct sums is isomorphic in the homotopy category to \({\mathcal C}^{\text{op}} \otimes^{\mathbb L} \widehat{{\mathcal D}}\). In particular there is a bijection between the set of homotopy classes \([\widehat {\mathcal C}, \widehat {\mathcal D}]_c\) and isomorphism classes in the homotopy category of \({\mathcal C} \otimes^{\mathbb L} {\mathcal D}^{\text{op}}\)-modules.
This theory not only provides the right framework to develop Morita theory for dg-algebras, it also allows the author to
(i) describe the homotopy groups of the classifying space of dg-categories in terms of Hochschild homology and the derived Picard group,
(ii) develop localization for dg-categories with respect to a set of morphisms in a given dg-category,
(iii) understand \(\mathbb R \operatorname{Hom}({\mathcal C}, {\mathcal D})\) when \({\mathcal C}\) and \({\mathcal D}\) are dg-categories of quasi-coherent or perfect complexes on certain schemes.

MSC:

18G55 Nonabelian homotopical algebra (MSC2010)
55U35 Abstract and axiomatic homotopy theory in algebraic topology
18G35 Chain complexes (category-theoretic aspects), dg categories
16D90 Module categories in associative algebras
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References:

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