Non-commutative deformations and quasi-coherent modules

TL;DR

This paper explores how certain algebraic structures called qcss presheaves of algebras can be deformed while preserving their module categories, linking algebraic and abelian deformations through explicit cohomological methods.

Contribution

It introduces a class of qcss presheaves for which well-behaved quasi-coherent module categories are defined and shows their stability under algebraic deformation, establishing a correspondence with abelian deformations.

Findings

Class of qcss presheaves with well-behaved Qch(A) defined.

Explicit parameterization of first order deformations via GS complex.

Equivalence of constructions with Toda's in the smooth case.

Abstract

We identify a class of "quasi-compact semi-separated" (qcss) twisted presheaves of algebras A for which well-behaved Grothendieck abelian categories of quasi-coherent modules Qch(A) are defined. This class is stable under algebraic deformation, giving rise to a 1-1 correspondence between algebraic deformations of A and abelian deformations of Qch(A). For a qcss presheaf A, we use the Gerstenhaber-Schack (GS) complex to explicitely parameterize the first order deformations. For a twisted presheaf A with central twists, we descibe an alternative category QPr(A) of quasi-coherent presheaves which is equivalent to Qch(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 O of the structure sheaf of a scheme X to a finite semi-separating open affine cover (for which we have an equivalence between Qch(O) and Qch(X)). Under a natural identification of Gerstenhaber-Schack cohomology of O and Hochschild cohomology of X, our construction is shown to be equivalent to Toda's construction in the smooth case.