Deformations of Quasicoherent Sheaves of Algebras

Gerstenhaber and Schack ([GS]) developed a deformation theory of presheaves of algebras on small categories. We translate their cohomological description to sheaf cohomology. More precisely, we describe the deformation space of (admissible) quasicoherent sheaves of algebras on a quasiprojective scheme $X$ in terms of sheaf cohomology on $X$ and $X\times X$. These results are applied to the study of deformations of the sheaf $D_X$ of differential operators on $X$. In particular, in case $X$ is a flag variety we show that any deformation of $D_X$, which is induced by a deformation of $\cO_X$, must be trivial. This result is used in [LR3], where we study the localization construction for quantum groups.