Non-commutative deformations and quasi-coherent modules

Abstract

We identify a class of quasi-compact semi-separated (qcss) twisted presheaves of algebras $$\mathcal {A}$$ for which well-behaved Grothendieck abelian categories of quasi-coherent modules $$\mathsf {Qch} (\mathcal {A})$$ are defined. This class is stable under algebraic deformation, giving rise to a 1–1 correspondence between algebraic deformations of $$\mathcal {A}$$ and abelian deformations of $$\mathsf {Qch} (\mathcal {A})$$ . For a qcss presheaf $$\mathcal {A}$$ , we use the Gerstenhaber–Schack (GS) complex to explicitly parameterize the first-order deformations. For a twisted presheaf $$\mathcal {A}$$ with central twists, we descibe an alternative category $$\mathsf {QPr} (\mathcal {A})$$ of quasi-coherent presheaves which is equivalent to $$\mathsf {Qch} (\mathcal {A})$$ , leading to an alternative, equivalent association of abelian deformations to GS cocycles of qcss presheaves of commutative algebras. Our construction applies to the restriction $$\mathcal {O}$$ of the structure sheaf of a scheme X to a finite semi-separating open affine cover (for which we have $$\mathsf {Qch} (\mathcal {O}) \cong \mathsf {Qch} (X)$$ ). Under a natural identification of GS cohomology of $$\mathcal {O}$$ and Hochschild cohomology of X, our construction is shown to be equivalent to Toda’s construction from Toda (J Differ Geom 81(1):197–224, 2009) in the smooth case.