A non-commutative deformation of a quadric surface is usually described by a three-dimensional cubic Artin-Schelter regular algebra. In this paper we show that for such an algebra its bounded derived category embeds into the bounded derived category of a commutative deformation of the Hilbert scheme of two points on the quadric. This is the second example in support of a conjecture by Orlov. Based on this example we formulate an infinitesimal version of the conjecture, and provide some evidence in the case of smooth projective surfaces.

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For any Koszul Artin-Schelter regular algebra A, we consider a version of the universal Hopf algebra aut(A) coacting on A, introduced by Manin. To study the representations (i.e. finite dimensional comodules) of this Hopf algebra, we use the Tannaka-Krein formalism. Specifically, we construct an explicit combinatorial rigid monoidal category U , equipped with a functor M to finite dimensional vector spaces such that aut(A) = coend U (M ). Using this pair (U , M ) we show that aut(A) is quasi-hereditary as a coalgebra and in addition is derived equivalent to the representation category of U .− → M (λ) and is thus in End U2 (M ). From the commutativity it follows that h induces f so γ(h) = f .Let I := λ∈Λ2−Λ1 I λ := λ∈Λ2−Λ1 Hom k (∇(λ), ∆(λ)) = ker β be the ideal constructed in Theorem 6.2.3.

For a smooth projective variety X with exceptional structure sheaf, and X [2] the Hilbert scheme of two points on X , we show that the Fourier-Mukai functor D b (X ) → D b (X [2] ) induced by the universal ideal sheaf is fully faithful, provided the dimension of X is at least 2. This fully faithfulness allows us to construct a spectral sequence relating the deformation theories of X and X [2] and to show that it degenerates at the second page, giving a Hochschild-Kostant-Rosenberg-type filtration on the Hochschild cohomology of X . These results generalise known results for surfaces due to Krug-Sosna, Fantechi and Hitchin. Finally, as a byproduct, we discover the following surprising phenomenon: for a smooth projective variety of dimension at least 3 with exceptional structure sheaf, it is rigid if and only if its Hilbert scheme of two points is rigid. This last fact contrasts drastically to the surface case: non-commutative deformations of a surface contribute to commutative deformations of its Hilbert square.

For an arbitrary finite-dimensional algebra A, we introduce a general approach to determining when its first Hochschild cohomology HH 1 (A), considered as a Lie algebra, is solvable. If A is moreover of tame or finite representation type, we are able to describe HH 1 (A) as the direct sum of a solvable Lie algebra and a sum of copies of sl2. We proceed to determine the exact number of such copies, and give an explicit formula for this number in terms of certain chains of Kronecker subquivers of the quiver of A. As a corollary, we obtain a precise answer to a question posed by Chaparro, Schroll and Solotar.