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 Mat. Sb., 2011, Volume 202, Number 4, Pages 31–64 (Mi msb7729)

Cohomological descent theory for a morphism of stacks and for equivariant derived categories

A. Elaginab

a A. A. Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences
b Laboratory of algebraic geometry and its applications, Higher School of Economics

Abstract: In the paper, we find necessary and sufficient conditions under which, if $X\to S$ is a morphism of algebraic varieties (or, in a more general case, of stacks), the derived category of $S$ can be recovered by using the tools of descent theory from the derived category of $X$. We show that for an action of a linearly reductive algebraic group $G$ on a scheme $X$ this result implies the equivalence of the derived category of $G$-equivariant sheaves on $X$ and the category of objects in the derived category of sheaves on $X$ with a given action of $G$ on each object.
Bibliography: 18 titles.

Keywords: derived categories, descent theory, algebraic variety.

DOI: https://doi.org/10.4213/sm7729

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English version:
Sbornik: Mathematics, 2011, 202:4, 495–526

Bibliographic databases:

Document Type: Article
UDC: 512.73
MSC: Primary 18E30, 18F20; Secondary 18A22, 18A25, 18A35, 18S40, 18D05, 18E10, 18G10

Citation: A. Elagin, “Cohomological descent theory for a morphism of stacks and for equivariant derived categories”, Mat. Sb., 202:4 (2011), 31–64; Sb. Math., 202:4 (2011), 495–526

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb7729
• https://doi.org/10.4213/sm7729
• http://mi.mathnet.ru/eng/msb/v202/i4/p31

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. Elagin, “Descent theory for semiorthogonal decompositions”, Sb. Math., 203:5 (2012), 645–676
2. M. Ballard, D. Favero, L. Katzarkov, “A category of kernels for equivariant factorizations and its implications for Hodge theory”, Publ. Math. Inst. Hautes Études Sci., 120 (2014), 1–111
3. Chen Xiao-Wu, “A note on separable functors and monads with an application to equivariant derived categories”, Abh. Math. Semin. Univ. Hambg, 85:1 (2015), 43–52
4. P. Seidel, “Picard-Lefschetz theory and dilating $\mathbb C^*$-actions”, J. Topol., 8:4 (2015), 1167–1201
5. A. Kuznetsov, A. Polishchuk, “Exceptional collections on isotropic Grassmannians”, J. J. Eur. Math. Soc. (JEMS), 18:3 (2016), 507–574
6. J. Hall, “The Balmer spectrum of a tame stack”, Ann. K-Theory, 1:3 (2016), 259–274
7. Kuznetsov A. Perry A., “Derived categories of cyclic covers and their branch divisors”, Sel. Math.-New Ser., 23:1 (2017), 389–423
8. Hirano Yu., “Equivalences of Derived Factorization Categories of Gauged Landau-Ginzburg Models”, Adv. Math., 306 (2017), 200–278
9. Novakovic S., “Tilting Objects on Some Global Quotient Stacks”, J. Commut. Algebr., 10:1 (2018), 107–137
10. Tabuada G., “Equivariant Noncommutative Motives”, Ann. K-Theory, 3:1 (2018), 125–156
11. Auel A. Bernardara M., “Semiorthogonal Decompositions and Birational Geometry of Del Pezzo Surfaces Over Arbitrary Fields”, Proc. London Math. Soc., 117:1 (2018), 1–64
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