论文标题
晶体棱镜：反射和衍射，现在和过去
Crystalline prisms: Reflections and diffractions, present and past
论文作者
论文摘要
令$ y/s $为$ p $ - $ p $ p $ torsion free $ p $ p $ p $ p $ addic正式方案，并带有frobenius升降机，然后让$ \ overline y/\ edline y/\ overline s $表示其还原模式$ $ p $。我们表明，$ \叠加y/s $的棱柱形位置上的晶体类别等于$ o_y $ - 模块的类别，具有可集成和准尼尔型$ p $ -p $ - 连接，并且相关的$ p $ -de rham复合物计算了这种晶体的共同体。更一般地，如果$ x $是$ \覆盖的y $ $ $ $，超过$ \叠加的s $，那么prismatic envelope $Δ_x（y）$ x $ in $ y $ in $ y $ in $ y $承认这样的$ p $ - connection，$ x/s $的类别是$ x $的类别。 Quasi-Nilpotent $ p $ - 连接，以及这种晶体的共同点再次由其$ P $ -DE RHAM综合体计算出来。我们还提供了``Prismatic Sen Operator''的几何结构。 $ x $。令人惊讶的是，这个综合体不是上述$ p $ p $ -de rham complextbut的减少模式，而不是它的``$α$ - 变形。 deligne-illusie分解定理。我们还解释了几位作者如何将希格斯字段，$ p $ - 连接和连接的早期作品放在棱柱上。
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 $Δ_X(Y)$ of $X$ in $Y$ admits such a $p$-connection, the category of prismatic crystals on $X/S$ is equivalent to the category of $O_ {Δ_X(Y)}$-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 $Δ_X(Y)$ 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 ``$α$-transform.'' As a consequence, we get a fairly explicit description of the action of the group scheme $G^γ$ on $RΓ(X, Ω^\bullet_{X/S})$, 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.
