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Derived categories and Grothendieck duality. (English) Zbl 1219.18006
Holm, Thorsten (ed.) et al., Triangulated categories. Based on a workshop, Leeds, UK, August 2006. Cambridge: Cambridge University Press (ISBN 978-0-521-74431-7/pbk). London Mathematical Society Lecture Note Series 375, 290-350 (2010).
Summary: We study dualizing complexes. The unusual feature is that we do not assume them to have bounded injective resolutions; we prove that the theory works just fine with no boundedness hypothesis. In the process we prove a number of new results about Grothendieck duality; one of the more striking is that, under relatively mild hypotheses on $$f:X\to Y$$, the functor $$f^!:{\mathbf D}(\text{Qcoh}/Y)\to{\mathbf D} (\text{Qcoh}/X)$$ takes pseudocoherent complexes to pseudocoherent complexes. The biggest innovation in our approach is that we systematically employ products in the category $${\mathbf D}(\text{Qcoh}/X)$$; the older treatments never ventured beyond coproducts.
In an appendix we include a proof that, if $$\mathcal T$$ is a stable homotopy category in the sense of M. Hovey, J. H. Palmieri and N. P. Strickland [Axiomatic stable homotopy theory, Mem. Am. Math. Soc. 610 (1997; Zbl 0881.55001)] in which the compact objects coincide with the strongly dualizable objects, then the compact objects can also be characterized as those objects such that tensoring with them respects products.
For the entire collection see [Zbl 1195.18001].

##### MSC:
 18E30 Derived categories, triangulated categories (MSC2010) 55U35 Abstract and axiomatic homotopy theory in algebraic topology 18G35 Chain complexes (category-theoretic aspects), dg categories 55U15 Chain complexes in algebraic topology 18-02 Research exposition (monographs, survey articles) pertaining to category theory