Notes on formal deformations of abelian categories (Q2869248)

The paper under review provides definitions and foundational results on deformations of abelian categories, with a view towards applications to non-commutative algebraic geometry. In particular, this material is used in other two works of the author, [Trans. Am. Math. Soc. 364, No. 12, 6279--6313 (2012; Zbl 1345.14009), and Int. Math. Res. Not. 2011, No. 17, 3983--4026 (2011; Zbl 1311.14003)]. In order to clarify the motivations, it is useful to start with a brief explanation of the set up of these two papers: as the general ideas are similar we can limit our discussion to the first one.NEWLINENEWLINE A \(\mathbb P^1\)-bundle \(\mathcal E\) on \(\mathbb P^1\) is determined by a rank two vector bundle \(E\) on \(\mathbb P^1\). Let \(\mathcal A\) be equal to \(\mathrm{Sym}^{\bullet} (E^{\ast})\). \(\mathcal A\) is a sheaf of commutative graded algebras, and the abelian category \(Coh(\mathcal E)\) of coherent sheaves on \(\mathcal E\) can be recovered as a localization of the category of graded modules over \(\mathcal A\). In [Zbl 1345.14009] the author defines the abelian category of coherent sheaves on a non-commutative \(\mathbb P^1\)-bundle over \(\mathbb P^1\) in an analogous fashion: that is, as a suitably localized abelian category of graded modules over a sheaf of non-commutative \(\mathbb Z\)-algebras \(\mathcal A\) over \(\mathbb P^1\), satisfying some natural additional conditions. One of the main results of [Zbl 1345.14009] is that if \(X\) is a commutative \(\mathbb P^1\)-bundle over \(\mathbb P^1\) (i.e. a Hirzebruch surface), all formal deformations of the abelian category \(Coh(X)\) are given by coherent sheaves on non commutative \(\mathbb P^1\) -bundles over \(\mathbb P^1\).NEWLINENEWLINEIn the paper under review the author provides the technical background required to make this claim precise. This builds on work of \textit{J. P. Jouanolou} [in: Semin. Geom. algebr. Bois--Marie 1965--66, SGA 5, Lect. Notes Math. 589, Expose No.V, 204--250 (1977; Zbl 0352.18019)]. Let \(R\) be a \(J\)-adic ring and let \(\mathcal C\) be a \(R/J\)-linear abelian category. The author defines a deformation of \(\mathcal C\) to be an abelian \(R\)-linear category \(\mathcal D\) such that subcategory of \(\mathcal D\) of objects annihilated by J is equal to \(\mathcal C\): additional properties are also required, including completeness and flatness. The first main result of the paper is an analogue of Grothendieck existence theorem: under mild conditions, the author shows that it is possible to compute Ext-groups in \(\mathcal D\) as a limit of calculations performed in \(R/J^n\) -linear subcategories of \(\mathcal D\). This can be found in Section 2.NEWLINENEWLINEIn Section 4 the author introduces strongly ample sequences in abelian categories. This is a refinement of previous notions due to Artin-Zhang and Polishchuck, and axiomatizes the properties of collections of the form \(\{\mathcal L^n \}_{n \in \mathbb Z}\), where \(\mathcal L\) is a very ample line bundle, in the category of coherent sheaves over a projective scheme. As before, let \(D\) be a formal deformation of the abelian category \(C\). The second main result of the paper gives conditions under which strongly ample sequences in \(C\) can be lifted to strongly ample sequences in \(D\). Both results play an important role in applications to non-commutative algebraic geometry.NEWLINENEWLINEFor the entire collection see [Zbl 1256.14001].