No abstract

Abstract. In this paper we develop Poisson geometry for non-commutative algebras. This generalizes the bi-symplectic geometry which was recently, and independently, introduced by Crawley-Boevey, Etingof and Ginzburg.Our (quasi-)Poisson brackets induce classical (quasi-)Poisson brackets on representation spaces. As an application we show that the moduli spaces of representations associated to the deformed multiplicative preprojective algebras recently introduced by Crawley-Boevey and Shaw carry a natural Poisson structure.

We give a sufficient condition for an Ext-finite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in existence of a strong generator. We prove that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, hence saturated. In contrast the similar category for a smooth compact analytic surface with no curves is not saturated.Theorem 1.1. Assume that X is a regular projective variety over a field k. Let D be the derived category of bounded coherent complexes on X. Then every contravariant cohomological functor of finite type on D is representable.This theorem was first announced in [8], but the proof in loc.cit works only for functors which are homologically bounded with respect to the standard t-structure. In the appendix to this paper we will give a short proof of a generalization of Theorem 1.1 which states that every contravariant cohomological functor of finite type on the derived category of perfect complexes on a (possibly singular) projective variety over a field is representable by a bounded complex of coherent sheaves. Theorem 1.1 is a motivation for the following definition [8]. Definition 1.2. Assume that D is Ext-finite, i.e. n dim Hom(A, B[n]) < ∞ for all A, B ∈ D. Then D is (right) saturated if every contravariant cohomological functor of finite type H : D → Vect(k) is representable. Saturated triangulated categories are significant for non-commutative algebraic geometry. It can be argued that any definition of a non-commutative proper scheme should 1991 Mathematics Subject Classification. Primary 18E30.

No abstract