Geodesic
Descent theory for semiorthogonal decompositions
Sbornik. Mathematics, Tome 203 (2012) no. 5, pp. 645-676 Cet article a éte moissonné depuis la source Math-Net.Ru

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We put forward a method for constructing semiorthogonal decompositions of the derived category of $G$-equivariant sheaves on a variety $X$ under the assumption that the derived category of sheaves on $X$ admits a semiorthogonal decomposition with components preserved by the action of the group $G$ on $X$. This method is used to obtain semiorthogonal decompositions of equivariant derived categories for projective bundles and blow-ups with a smooth centre as well as for varieties with a full exceptional collection preserved by the group action. Our main technical tool is descent theory for derived categories. Bibliography: 12 titles.
Keywords: derived category, descent theory, algebraic variety.
Mots-clés : semiorthogonal decomposition 
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A. Elagin. Descent theory for semiorthogonal decompositions. Sbornik. Mathematics, Tome 203 (2012) no. 5, pp. 645-676. http://geodesic.mathdoc.fr/item/SM_2012_203_5_a1/

[1] A. D. Elagin, “Semiorthogonal decompositions of derived categories of equivariant coherent sheaves”, Izv. Math., 73:5 (2009), 893–920 | DOI | MR | Zbl

[2] A. I. Bondal, M. Van den Bergh, “Generators and representability of functors in commutative and noncommutative geometry”, Mosc. Math. J., 3:1 (2003), 1–36 | MR | Zbl

[3] N. Spaltenstein, “Resolutions of unbounded complexes”, Compositio Math., 65:2 (1988), 121–154 | MR | Zbl

[4] G. Laumon, L. Moret-Bailly, Champs algébriques, Ergeb. Math. Grenzgeb. (3), 39, Springer-Verlag, Berlin, 2000 | MR | Zbl

[5] M. Kashiwara, P. Shapira, Categories and sheaves, Grundlehren Math. Wiss., 332, Springer-Verlag, Berlin, 2006 | MR | Zbl

[6] M. Barr, Ch. Wells, “Toposes, triples and theories”, Repr. Theory Appl. Categ., 12 (2005), 1–288 | MR | Zbl

[7] S. Mac Lane, Categories for the working mathematician, Grad. Texts in Math., 5, Springer-Verlag, New York, 1998 | MR | Zbl

[8] A. D. Elagin, “Cohomological descent theory for a morphism of stacks and for equivariant derived categories”, Sb. Math., 202:4 (2011), 495–526 | DOI | MR | Zbl

[9] D. Mumford, J. Fogarty, Geometric invariant theory, Ergeb. Math. Grenzgeb., 34, Springer-Verlag, Berlin–Heidelberg–New York, 1982 | MR | Zbl

[10] A. Kuznetsov, “Base change for semiorthogonal decompositions”, Compos. Math., 147:3 (2011), 852–876 | DOI | MR | Zbl

[11] A. D. Elagin, “Equivariant derived category of bundles of projective spaces”, Proc. Steklov Inst. Math., 264:1 (2009), 56–61 | DOI | MR

[12] D. O. Orlov, “Projective bundles, monoidal transformations, and derived categories of coherent sheaves”, Russian Acad. Sci. Izv. Math., 41:1 (1993), 133–141 | DOI | MR | Zbl