Michel Van den Bergh

Non-commutative Crepant Resolutions

We introduce the notion of a “non-commutative crepant” resolution of a singularity and show that it exists in certain cases. We also give some evidence for an extension of a conjecture by Bondal and Orlov, stating that different crepant resolutions of a Gorenstein singularity have the same derived category.

A relation between Hochschild homology and cohomology for Gorenstein rings

The paper [5] contains an error in the sense that Theorem 1 (the “duality theorem”) is false in the generality stated. As a result the same is true for its corollaries: Proposition 3 and Corollary 6. The main conclusion, which is an affirmative answer to a question by Patrick Polo, remains valid however (see below). That Theorem 1 is false as stated was pointed out in [2]. In general it can be seen as follows. If the conclusion of Theorem 1 is true, then the Hochschild dimension (the cohomological dimension of HH∗) of the ring A is finite. So Theorem 1 must be false for every ring of infinite Hochschild dimension, and hence in particular for every ring of infinite global dimension. Thus to save Theorem 1 we must assume that A has finite Hochschild dimension (let us say that A is smooth in this case). It is easy to see that in that case the proof becomes valid. The smoothness hypothesis is automatically satisfied in Proposition 2 (see [4]) but it must be added in Proposition 3 and Corollary 6. The following lemma shows that smoothness is a reasonable condition.

Three-dimensional flops and noncommutative rings

Limburgs Univ Ctr, Dept Math, B-3590 Diepenbeek, Belgium.Van den Bergh, M, Limburgs Univ Ctr, Dept Math, Univ Campus, B-3590 Diepenbeek, Belgium.vdbergh@luc.ac.be

DEFORMED CALABI-YAU COMPLETIONS

Abstract We define and investigate deformed n-Calabi–Yau completions of homologically smooth differential graded (= dg) categories. Important examples are: deformed preprojective algebras of connected non-Dynkin quivers, Ginzburg dg algebras associated to quivers with potentials and dg categories associated to the category of coherent sheaves on the canonical bundle of a smooth variety. We show that deformed Calabi–Yau completions do have the Calabi–Yau property and that their construction is compatible with derived equivalences and with localizations. In particular, Ginzburg dg algebras have the Calabi–Yau property. We show that deformed 3-Calabi–Yau completions of algebras of global dimension at most 2 are quasi-isomorphic to Ginzburg dg algebras and apply this to the study of cluster-tilted algebras and to the construction of derived equivalences associated to mutations of quivers with potentials. In the appendix, Michel Van den Bergh uses non-commutative differential geometry to give an alternative proof of the fact that Ginzburg dg algebras have the Calabi–Yau property.

Double Poisson algebras

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.

Semi-invariants of quivers for arbitrary dimension vectors

Univ Bristol, Dept Math, Bristol BS8 1TH, Avon, England. Limburgs Univ Ctr, Dept WNI, B-3590 Diepenbeek, Belgium.

Simplicity of rings of differential operators in prime characteristic

Let

Deformation theory of abelian categories

W

Hochschild cohomology and Atiyah classes

be a finite dimensional representation of a linearly reductive group

A relation between a conjecture of Kac and the structure of the Hall algebra

G