## 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

### MSC:

 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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### References:

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