Vadim Vologodsky: Sheared prismatization

Abstract: (The talk is based on a joint work in progress with
Bhargav Bhatt, Artem Kanaev, Akhil Mathew, and Mingjia Zhang.) Let X be a smooth p-adic formal scheme and let X^{dR} be its de Rham stack. The category of quasi-coherent sheaves on the latter is equivalent to the category of p-complete D-modules M such that the p-curvature of M/p is locally nilpotent.
Bhatt, Lurie, and Drinfeld constructed a natural deformation of its de Rham stack X^{dR} parametrized by a certain stack Z_p^\Delta. On the other hand, one has a stack X^{\hat{dR}}, whose category of quasi-coherent sheaves is equivalent to the category of all p-complete D-modules.
I will construct a torsor Z_p^{\hat{\Delta}} over Z_p^\Delta and a deformation of X^{\hat{dR}}nparametrized by Z_p^{\hat{\Delta}}. In particular, this construction determines a canonical G_m-gerbe over the cotangent bundle to X whose mod p fiber is represented by the Azumaya algebra of differential operators and whose generic fiber is related to the Simpson gerbe.

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Gonzalez Eric Harris Michelle Miller Jeffrey