Tilting sheaves in representation theory of algebras

Abstract

For a nonsingular weighted projective varietyX we introduce the notion of a tilting sheaf onX. We characterize tilting sheaves as those sheavesM ∈ coh (X) that lead to an equivalence φ: D^b(coh (X))→ D^b(mod (B)) with φ(M)=B, whereB is the endomorphism algebra ofM. An induced comparison theorem interrelates directly certain subcategories of coh (X) and mod(B), respectively. For a weighted projective spaceX, these subcategories control vector bundles up to twist and, ifM is itself a bundle, also all coherent sheaves.

There are self-equivalences of D^b(mod (B)), induced by the twist in the category of sheaves, that can be considered as a generalization of Coxeter functors. In special situations, these functors are related to the Auslander-Reiten translation and serve as “higher” Auslander-Reiten functors.

For the projectiven-space, reduction procedures lead to up to twist descriptions of coherent sheaves by modules over algebrasS[t, n] of global dimensionn−t,0≤t≤n−1. D^b(coh (P _n )) is equivalent to the homotopy category of bounded complexes of suitable modules over each of these algebras.