Hochschild cohomology of noncommutative planes and quadrics (Q2327777)

\textit{M. Artin} and \textit{J. J. Zhang} [Adv. Math. 109, No. 2, 228--287 (1994; Zbl 0833.14002)] defined noncommutative algebraic geometry by identifying noncommutative projective varieties by the abelian categories of coherent sheaves on not necessarily commutative graded algebras. The properties of the analogues of commutative polynomial rings have been axiomatised by \textit{M. Artin} and \textit{W. F. Schelter} [ibid. 66, 171--216 (1987; Zbl 0633.16001)], and a full classification of 3-dimensional Artin-Schelter regular algebras is given by \textit{M. Artin} et al. [Invent. Math. 106, No. 2, 335--388 (1991; Zbl 0763.14001)]. This classification is interpreted as the classification of noncommutative planes and noncommutative quadrics. There are precisely two types of graded algebras in this case, distinguished by their Hilbert series and corresponding to the noncommutative analogues of \(\mathbb{P}^2\) or \(\mathbb{P}^1\times\mathbb{P}^1\). \textit{W. Lowen} and \textit{M. van den Bergh} [Trans. Am. Math. Soc. 358, No. 12, 5441--5483 (2006; Zbl 1113.13009)] gave a deformation theory for abelian categories, generalizing that of associative algebras. The infinitesimal deformations are governed by a Hochschild cohomology for abelian categories, and this Hochschild cohomology describes the infinitesimal deformations as an abelian category. In this article, the computations of the Hochschild cohomology of the abelian category of coherent sheaves on \(\mathbb{P}^1\times\mathbb{P}^1\) proves that the classification of noncommutative quadrics is incomplete if one uses only graded algebras. There should be \(3\) degrees of freedom, but the second Hochschild cohomology of coherent sheaves on graded algebras have at most dimension \(2\). \textit{M. Van den Bergh} [Int. Math. Res. Not. 2011, No. 17, 3983--4026 (2011; Zbl 1311.14003)] generalised the construction of noncommutative quadrics to \(\mathbb{Z}\)-algebras and gave a classification of these in terms of geometric and linear data. For noncommutative planes, the classification using \(\mathbb{Z}\)-algebras gives a more streamlined approach that is used in this article. Here, the Hochschild cohomology of all noncommutative planes and quadrics is described using geometric techniques. From given tables of Hochschild cohomology of noncommutative planes and quadrics, it appears that these have less infinitesimal symmetries, and less infinitesimal deformations explaining the dimension-drop. An explicit explanation of the behaviour of the dimension-drop in all cases is eventually given. The author recalls the notions of Hochschild cohomology of algebras, varieties and abelian categories, and gives the properties necessary for the present article. The exceptional objects giving the translation between smooth projective varieties and associative algebras are recalled, and so are the technicalities regarding \(\mathbb{Z}\)-algebras needed for the classification and definition of noncommutative planes and quadrics. The paper contains an overview of a not so well-known classification of base loci of pencils of quadrics that arise as the point schemes of noncommutative quadrics. Based on the Lefschetz trace formula for Hochschild cohomology, the method of computing is given, leading to the description of the first Hochschild cohomology of a finite dimensional algebra as the Lie algebra of its outer automorphisms. It is explained how this group is related to the automorphism groups of the corresponding \(\mathbb{Z}\)-algebra, and by this how the geometric classification of noncommutative planes (quadrics) given by \textit{A. I. Bondal} and \textit{A. E. Polishchuk} [Russ. Acad. Sci., Izv., Math. 42, No. 2, 219--260 (1993; Zbl 0847.16010); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 57, No. 2, 3--50 (1993)] (\textit{M. Van den Bergh} [Int. Math. Res. Not. 2011, No. 17, 3983--4026 (2011; Zbl 1311.14003)]) can be reduced to a purely geometric question regarding automorphism groups of certain curves (with bad properties) of arithmetic genus \(1\). The article contains descriptions of the automorphism groups necessary for the above classification, and for noncommutative quadrics this description is new. Also, a final comment on another class of finite-dimensional algebras arising from algebraic geometry where the Hochschild cohomology is completely known is given. For weighted projective lines it turns out that there is no dependence on the parameters, i.e. the location of the stacky points. The article is very well written, and contains explicit computations and examples. The resulting tables are of great value, and a lot of important concepts are defined and explained: First of all, \(k\) is an algebraically closed field of \(\operatorname{char}k\neq 2,3\). The definition of the Hochshild cohomology of an abelian category \(\mathcal{A}\) is given by \(\operatorname{HH}^\bullet_{\text{ab}}(\mathcal{A}):=\operatorname{HH}^\bullet_{\text{ab}}(\text{Inj Ind }\mathcal{A})\) where \(\text{Inj Ind }\mathcal{A}\) is the dg category associated to the \(k\)-linear subcategory of injective objects of the Ind-completion of \(\mathcal A\). This definition extends the definition of Hochschild cohomology in a nontrivial way, and via exceptional objects, one sees that on the algebraic level, the Hochschild cohomology depends heavily on the relations in the quiver of the algebra in question. This is in some sense the building blocks of the main results of the article. Also, the category of Artin-Schelter algebras needs to be enhanced, and that is done by the definition of Artin-Schelter regular \(\mathbb{Z}\)-algebras. A \(\mathbb{Z}\)-algebra is a non-unital associative algebra \(A=\oplus_{i,j\in\mathbb{Z}}A_{ij}\) for which \(A_{i,j}\) is a finite-dimensional vector space, the subalgebra \(A_{i,i}\) is isomorphic to \(k\) for all \(i\in\mathbb{Z}\), and \(A_{i,j}=0\) if \(ji\). Also, the multiplication takes the grading into account. The classification of noncommutative quadrics applies the notion of Segre symbols, which is an invariant of linear transformations, and the first Hochschild cohomology as Lie algebra.