Picard groups of derived categories. (English) Zbl 1023.18011
In [A. Zimmermann, CMS Conf. Proc. 18, 721-749 (1996; Zbl 0855.16015)] and R. Rouquier and A. Zimmermann [“Picard groups for derived module categories”, Proc. Lond. Math. Soc. 87, 197-225 (2003; Zbl 1058.18007)] Rouquier and the reviewer and independently A. Yekutieli [J. Lond. Math. Soc., II. Ser. 60, 723-746 (1999; Zbl 0954.16006)] defined the group of standard self-equivalences of the derived category of an algebra. One of the statements proved in these papers in varying generality was the fact that for a commutative indecomposable ring \(R\) any self-equivalence of the derived category of bounded complexes of \(R\)-modules is given by a composition of a shift in degree and a Morita self-equivalence. The author proves in the paper under review basically the same kind of statement replacing a commutative ring by a commutative unital ringed Grothendieck topos with enough points. The method of proof is similar to the one in [Rouquier and Zimmermann, loc. cit.], though, since the setting is much more general, many technical difficulties are considerably harder in the paper under review.
Reviewer: Alexander Zimmermann (Amiens)
MSC:
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 14C22 | Picard groups |
| 14F20 | Étale and other Grothendieck topologies and (co)homologies |
| 16D90 | Module categories in associative algebras |
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