Twisting the Infinitesimal Site

We classify twistings of Grothendieck's differential operators on a smooth variety $X$ in prime characteristic $p$. We prove that isomorphism classes of twistings are in bijection with $H<^>{2}(X,\mathbb{Z}_{p}(1))$, the degree 2, weight 1 syntomic cohomology of $X$. We also discuss the relationship between twistings of crystalline and Grothendieck differential operators. Twistings in mixed characteristic are also analyzed.