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On the derived category of coherent sheaves on Grassmann manifolds. (English. Russian original) Zbl 0564.14023
Math. USSR, Izv. 24, 183-192 (1985); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 48, No. 1, 192-202 (1984).
Let \({\mathbb{C}}\) be the field of complex numbers (or any algebraically closed field of zero characteristic), V an n-dimensional \({\mathbb{C}}\)-space and \(G=G(k,V)\) the Grassmannian of k-dimensional subspaces of V. The author describes the derived category \(D^ b_{coh}(G)\) of coherent sheaves on G. The main results asserts that \(D^ b_{coh}(G)\) is equivalent with the triangulated category of homotopical category of bounded complexes of graduated \(\bigwedge (V^*)\)-modules, consisting of finite coproducts of modules \(M_{\alpha}\). The modules \(M_{\alpha}\) are canonical associated to Young’s diagrams \(\alpha\) with at most k-rows and most n-k-columns.
Reviewer: N.Popescu

MSC:
14M15 Grassmannians, Schubert varieties, flag manifolds
18E30 Derived categories, triangulated categories (MSC2010)
15A75 Exterior algebra, Grassmann algebras
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
32L05 Holomorphic bundles and generalizations
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