Non-commutative deformations and quasi-coherent modules

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 GerstenhaberSchack (GS) complex to explicitly 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 Qch(O) ∼ = Qch(X )). Under a natural identification of GS cohomology of O and

Deformations of categories of coherent sheaves via quivers with relations

“…This shows that the deformation theory of Mod(O X ), Qcoh(X) and coh(X) can be understood as an amalgamation of quantizations of algebraic Poisson structures on X, classical commutative deformations of X and deformations of the gerbe structure of O X (see § 3). Further aspects of this deformation theory were studied for example in [BBP07,DHL22,DLL17,LL19].…”