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

This article is cited in 11 scientific papers (total in 11 papers)

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

Full text: PDF file (678 kB)
References: PDF file   HTML file

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
Received: 27.04.2010 and 06.10.2010

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

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. Elagin, “Descent theory for semiorthogonal decompositions”, Sb. Math., 203:5 (2012), 645–676  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    4. P. Seidel, “Picard-Lefschetz theory and dilating $\mathbb C^*$-actions”, J. Topol., 8:4 (2015), 1167–1201  crossref  mathscinet  zmath  isi  scopus
    5. A. Kuznetsov, A. Polishchuk, “Exceptional collections on isotropic Grassmannians”, J. J. Eur. Math. Soc. (JEMS), 18:3 (2016), 507–574  crossref  mathscinet  zmath  isi  scopus
    6. J. Hall, “The Balmer spectrum of a tame stack”, Ann. K-Theory, 1:3 (2016), 259–274  crossref  mathscinet  zmath  isi
    7. Kuznetsov A. Perry A., “Derived categories of cyclic covers and their branch divisors”, Sel. Math.-New Ser., 23:1 (2017), 389–423  crossref  mathscinet  zmath  isi  scopus
    8. Hirano Yu., “Equivalences of Derived Factorization Categories of Gauged Landau-Ginzburg Models”, Adv. Math., 306 (2017), 200–278  crossref  mathscinet  zmath  isi  elib  scopus
    9. Novakovic S., “Tilting Objects on Some Global Quotient Stacks”, J. Commut. Algebr., 10:1 (2018), 107–137  crossref  mathscinet  zmath  isi  scopus
    10. Tabuada G., “Equivariant Noncommutative Motives”, Ann. K-Theory, 3:1 (2018), 125–156  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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