Crystalline prisms: Reflections and diffractions, present and past
Let $Y/S$ be a $p$-completely smooth morphism of $p$-torsion free $p$-adic formal schemes endowed with a Frobenius lift, and let $\overline Y/\overline S$ denote its reduction modulo $p$. We show that the category of crystals on the prismatic site of $\overline Y/S$ is equivalent to the category of $O_Y$-modules with integrable and quasi-nilpotent $p$-connection, and that the cohomology of such a crystal is computed by the associated $p$-de Rham complex. More generally, if $X$ is a closed subscheme of $\overline Y$, smooth over $\overline S$, then the prismatic envelope $\Delta_X(Y)$ of $X$ in $Y$ admits such a $p$-connection, the category of prismatic crystals on $X/S$ is equivalent to the category of $O_ {\Delta_X(Y)}$-modules with compatible integrable and quasi-nilpotent $p$-connection, and the cohomology of such a crystal is again computed by its $p$-de Rham complex. We also give a geometric construction of the ``prismatic Sen operator.'' Namely, we show that a lifting of $X$ (mod $p^2$) in $Y$ defines a vector field on the reduction modulo $p$ of $\Delta_X(Y)$ and on a ``diffracted'' Higgs complex which calculates the mod $p$ prismatic and de Rham cohomologies of $X$. Surprisingly, this complex is not the reduction modulo $p$ of the afore-mentioned $p$-de Rham complexbut is rather its ``$\alpha$-transform.'' As a consequence, we get a fairly explicit description of the action of the group scheme $G^\gamma$ on $R\Gamma(X, \Omega^\bullet_{X/S})$, Drinfeld's strengthening of the Deligne-Illusie decomposition theorem. We also explain how earlier work by several authors relating Higgs fields, $p$-connections, and connections can be placed in the prismatic context.