Tilting sheaves in representation theory of algebras (Q1100260)

For a nonsingular weighted projective variety X (resp. finite dimensional algebra B) denote by D b(coh(X)) (resp. D b(mod(B))) the derived category of bounded complexes over the category coh(X) of coherent sheaves on X (resp. the category mod(B) of finite dimensional B-modules). A tilting sheaf on X is a coherent sheaf M without self-extensions that generates D b(coh(X)) as a triangulated category and whose endomorphism algebra B has finite global dimension. The main result shows that \(M\in coh(X)\) with \(End(M)=B\) is a tilting sheaf if and only if there exists an equivalence F: D b(coh(X))\(\to D\) b(mod(B)) of triangulated categories such that \(F(M)=B\). Hence some equivalences between certain subcategories of coh(X) and mod(B) are established. In the paper, the tilting sheaves on the weighted projective space \(X=P_ n(p)\) and the ordinary projective n-space \(P_ n\) are particularly investigated. Moreover, the tilting procedures from modules to modules are also considered.