  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.
