The Nori-Hilbert scheme of an associative algebra A, which is a generalization of the usual Hilbert scheme of n-points of a commutative ring, can be considered as a litmus paper to test "smoothness" conditions satisfied by an associative algebra of low cohomological dimension.
Indeed it is a classical result that the usual Hilbert scheme of n-points is smooth when computed over smooth quasi-projective curves or surfaces.
In this talks we will give some recents results concerning this problem, namely:
- the above scheme is smooth when the second Ext(M,M)=0 for all finite dimensional A-modules (this confirms what was known for finite dimensional algebras)
- the above scheme is in general not smooth for 2-Calabi Yau algebras (we will provide a counterexample by means of the group algebras of the fundamental group of topological surfaces of genus greater than one).
We will give a detailed description of the local geometry of the above in terms of Hochschild cohomology of A.
Joint works with Alessandro Ardizzoni, Raf Bocklandt and Federica Galluzzi