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For a smooth variety $X$ over an algebraically closed field of characteristic$p$, to a differential 1-form $\alpha$ on the Frobenius twist $X^{(1)}$ one canassociate an Azumaya algebra $\mathcal D_{X,\alpha}$, defined as a certaincentral reduction of the algebra $\mathcal D_X$ of "crystalline differentialoperators" on $X$. For a resolution of singularities $\pi:X\to Y$ of an affinevariety $Y$, we study for which $\alpha$ does the class $[\mathcalD_{X,\alpha}]$ in the Brauer group $\mathrm{Br}(X^{(1)})$ descend to $Y^{(1)}$.In the case when $X$ is symplectic, this question is related to Fedosovquantizations in characteristic $p$ and the construction of non-commutativeresolutions of $Y$. We prove that the classes $[\mathcal D_{X,\alpha}]$ descend\'etale locally for all $\alpha$ if $\mathcal O_Y\simeq \pi_* \mathcal O_X$ and$R^{1,2}\pi_*\mathcal O_X =0$. We also define a certain class of resolutionswhich we call resolutions with conical slices, and prove that for a generalreduction of a resolution with conical slices in characteristic $0$ to analgebraically closed field of characteristic $p$ classes $[\mathcalD_{X,\alpha}]$ descend to $Y^{(1)}$ globally for all $\alpha$. Finally we givesome examples, in particular we show that Slodowy slices, Nakajima quivervarieties and hypertoric varieties are resolutions with conical slices.

Resolutions with conical slices and descent for the Brauer group classes  of certain central reductions of differential operators in characteristic $p$