On the derived categories of coherent sheaves on some homogeneous spaces. (English) Zbl 0651.18008

We quote in part from the paper: The derived category of coherent sheaves on the projective space P n has been shown to be equivalent (as a triangulated category) to the homotopy category of finite complexes of sheaves, consisting of finite direct sums of \({\mathcal O}(-i),\) \(i=0,1,...,n\). For a preadditive category \({\mathcal A}\) the author forms a triangulated category Tr(\({\mathcal A})\) which is “generated freely” by \({\mathcal A}\). Its objects are finite complexes, consisting of finite formal direct sums of objects of \({\mathcal A}\), and morphisms are homotopy classes of morphisms of complexes. The author then represents in the form Tr(\({\mathcal A})\) the derived categories of coherent sheaves on flag varieties and quadrics, and also the derived categories of finite- dimensional representations of parabolic subgroups of GL(n,\({\mathbb{C}})\).
Reviewer: P.Cherenack


18E30 Derived categories, triangulated categories (MSC2010)
14M17 Homogeneous spaces and generalizations
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14F20 Étale and other Grothendieck topologies and (co)homologies
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
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[1] Backelin, J.: On the rates of growth of the homologies of Veronese subrings. (Lect. Notes Math., vol. 1183, p. 79-100). Berlin Heidelberg New York: Springer 1986 · Zbl 0588.13011
[2] Beilinson, A.A.: Coherent sheaves onP n and problems of linear algebra. Funkcionalniy analiz i ego pril.12, 68-69 (1978) (in Russian)
[3] Beilinson, A.A.: The derived category of coherent sheaves onP n . In: Questions of group theory and homological algebra, vol. 2, Aroslavl state univ., 1979, p. 42-54 (in Russian).
[4] Beilinson, A.A., Bernstein, I.N., Deligne, P.: Faisceaux pervers. Astérisque100, 1981
[5] Bernstein, I.N., Gelfand, I.M., Gelfand, S.I.: Algebraic vector bundles onP n and problems of linear algebra. Funkcionalniy analiz i ego pril.12, 66-67 (1978) (in Russian) · Zbl 0402.14005
[6] Brauer, R., Weyl, H.: Spinors inN dimensions. Am. J. Math.57, 425-449 (1935) · Zbl 0011.24401
[7] Drezet, J.: M. Fibres exceptionels et suite spectrale de Beilinson generalisee surP 2(C). Math. Ann.275, 25-48 (1986) · Zbl 0578.14013
[8] Gorodentsev, A.L., Rudakov, A.N.: Exceptional bundles on projective spaces. Duke Math. J. (in press) · Zbl 0646.14014
[9] Hartshorne, R.: Algebraic geometry. Berlin Heidelberg New York: Springer 1977 · Zbl 0367.14001
[10] Hartshorne R.: Residues and duality. (Lect. Notes Math. vol. 20) Berlin Heidelberg New York: Springer 1966 · Zbl 0212.26101
[11] Kapranov, M.M.: On the derived category of coherent sheaves on Grassmann varieties. USSR Math. Izvestija48, 192-202 (1984) · Zbl 0564.14023
[12] Kapranov, M.M.: The derived category of coherent sheaves on a quadric. Funkcionalniy analiz i ego pril.20, 67 (1986) (in Russian) · Zbl 0607.18004
[13] Kostant, B.: Lie algebra cohomology and the generalised Borel-Weil theorem. Ann. Math.74, 329-387 (1961) · Zbl 0134.03501
[14] Lofwall, C.: On the subalgebra generated by one-dimensional elements in the Yoneda Ext-algebia. (Lect. Notes Math. vol. 1183, p. 291-338). Berlin Heidelberg New York: Springer 1986
[15] Macdonald, I.: Symmetric functions and Hall polynomials. Oxford: Clarendon Press, 1979 · Zbl 0487.20007
[16] McLane, S.: Homology. Berlin Heidelberg New York: Springer, 1963
[17] Okonek, C., Schneider, M., Spindler, H.: Vector bundles on complex projective spaces. Basel Boston Birkhäuser, 1980 · Zbl 0438.32016
[18] Priddy, S.: Koszul complexes. Trans. Am. Math. Soc.152, 1970 · Zbl 0199.33301
[19] Swan, R.G.:K-theory of quadric hypersurfaces. Ann. Math.121, 113-153 (1985) · Zbl 0601.14009
[20] Tate, J.: Homology of Noetherian rings and local rings. Ill. J. Math.1, 14-27 (1957) · Zbl 0079.05501
[21] Verdier, J.-L.: Categories derivees. (Lect. Notes Math. 569, p. 262-311). Berlin Heidelberg New York: Springer 1977
[22] Weyl, H.: Classical groups. Princeton Princeton University Press, 1941 · JFM 67.0078.02
[23] Whitehead, G.W.: Recent advances in homotopy theory. Am. Math. Soc. regional conference series in Math., vol. 5, 1970 · Zbl 0217.48601
[24] Cohen, J.M.: The decomposition of stable homotopy. Ann. Math.87, 305-320 (1968) · Zbl 0162.55102
[25] O’Brien N., Toledo D., Tong Y.L.: The trace and characteristic classes of coherent sheaves. Am. J. Math.103, 225-252 (1981) · Zbl 0473.14008
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