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 |