论文标题
一般基础上的晶体共同体
Crystalline cohomology over general bases
论文作者
论文摘要
在Berthelot的想法的基础上，我们为任何环$ a $ $ a $ $ \ mathbf {z} $ - flat $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $。对于平滑的$ a $ -scheme $ y $和一个封闭的亚cheme $ x $ $ y $，$η$扩展到$ i_0 \ i_0 \ mathscr {o} _x $，a（quasi-corherent）crystal $ \ mathscr {f} $ on $（x/a） $ a $ a $ - 线性连接在某个完成$ d_ {y，η}（x）^{\ wedge} $（称为“ pd-adic”）的划分电源信封的$ d_ {y，η}（x）$ y $ y $ y $ y $ x $的$ y $（带有divield withe wither结构$δ$δ$δ$δ$δ$δ$δ$δ$δ$δ$ 我们的主要结果是基于Bhatt和de Jong的想法，以$ \ mathbf {z}/（p^e）$ - $ - 方案（其中PD-ADIC完成无效）是$ {\ rm {rm {rm {rm {rm {rm {rm {rm {rm {rm {rm {rm {rm}}γ（x/x/a） Pd ofed完成的de rham复合物的过度酒精学$ d_ {y，η}（x）^{\ wedge} $关联到$ \ mathscr {f} $。通过相同方法的变体，我们获得了复杂$ \ Mathscr {f} \ wideHat {\ otimes} \wideHatΩ^*_ {d_ {d_ {y，η}（x）^{x） Čech-Alexander建筑。 当$ \ mathscr {f} = \ mathscr {o} _ {x/a} $时，我们的比较定理意味着，在$ x $的$ a $ a $ a-modules的派生类别中，$ x $，pd-adic的pd-adic完成$ω^*_ {d_ {d_ {d_ {y {y {y} $ nouth houth，n party n partery nouth untion。超过$ \ mathbf {q} $ - 代数$ a $，因此PD-Adic完成成为理想的成就，这是Hartshorne的结果。
Building on ideas of Berthelot, we develop a crystalline cohomology formalism over divided power rings $(A, I_0, η)$ for any ring $A$, allowing $\mathbf{Z}$-flat $A$. For a smooth $A$-scheme $Y$ and a closed subscheme $X$ of $Y$ for which $η$ extends to $I_0 \mathscr{O}_X$, a (quasi-coherent) crystal $\mathscr{F}$ on $(X/A)_{\rm{cris}}$ is equivalent to a specific type of module with integrable $A$-linear connection over a certain completion $D_{Y,η}(X)^{\wedge}$ (called "pd-adic") of the divided power envelope $D_{Y,η}(X)$ of $Y$ along $X$ (with divided power structure $δ$) Our main result, building on ideas of Bhatt and de Jong for $\mathbf{Z}/(p^e)$-schemes (where pd-adic completion has no effect), is a natural isomorphism between ${\rm{R}}Γ((X/A)_{\rm{cris}}, \mathscr{F})$ and the Zariski hypercohomology of the pd-adically completed de Rham complex $\mathscr{F} \widehat{\otimes} \widehatΩ^*_{D_{Y,η}(X)^{\wedge}/A,δ}$ arising from the module with integrable connection over $D_{Y,η}(X)^{\wedge}$ associated to $\mathscr{F}$. By a variant of the same methods, we obtain a representative of the complex $\mathscr{F} \widehat{\otimes} \widehatΩ^*_{D_{Y,η}(X)^{\wedge}/A,δ}$ in the derived category of sheaves of $A$-modules on $X$ in terms of a Čech-Alexander construction. When $\mathscr{F}=\mathscr{O}_{X/A}$, our comparison theorem implies that in the derived category of sheaves of $A$-modules on $X$, the pd-adic completion of $Ω^*_{D_{Y,η}(X)/A,δ}$ functorially depends only on $X$. Over $\mathbf{Q}$-algebras $A$, so pd-adic completion becomes ideal-adic completion, this recovers a result of Hartshorne.
