论文标题
理性$ p $ -adic hodge理论$ d $ -de rham-proper堆栈
Rational $p$-adic Hodge theory for $d$-de Rham-proper stacks
论文作者
论文摘要
在这篇后续文件中，我们表明，$ \ Mathcal O_K $上的光滑Hodge-Proper堆栈是$ \ Mathbb Q_p $ - locally acyclicl：即，代数和Raynaud通用纤维的自然图是o \ Mathbb q_p Q_p $ - Q_p Q_P $ - 物种学。这建立了我们以前的工作中做出的一般猜想的$ \ mathbb q_p $。作为推论，我们会发现，如果超过$ k $的平滑Artin堆栈具有$ \ Mathcal O_K $的光滑Hodge-Proper型号，则其$ \ Mathbb Q_P $ - Étale共同体是一种结晶的Galois表示。然后，我们还建立了上述结果的截断版本，以更普通的$ d $ d $ de $ de rham-proper堆栈的更通用设置，而不是$ \ mathcal o_k $：在这里，我们只需要$ \ \ \ \ nathcal o_k $，在这里只需要首先$ d $ d $ d $ d $ de rham groun。作为一种应用程序，我们针对raynaud通用纤维的o <tale $ \ mathbb q_p $ \ mathbb q_p $ - 在$ \ mathcal o_k $的情况下，在cohen--macauley在架构设置的情况下，提取了一定的纯度类型语句，以及前几个odtale od-macauley组的结晶度。
In this follow-up paper we show that smooth Hodge-proper stacks over $\mathcal O_K$ are $\mathbb Q_p$-locally acyclic: namely the natural map between étale $\mathbb Q_p$-cohomology of the algebraic and Raynaud generic fibers is an equivalence. This establishes the $\mathbb Q_p$-case of general conjectures made in our previous work. As a corollary, we get that if a smooth Artin stack over $K$ has a smooth Hodge-proper model over $\mathcal O_K$, its $\mathbb Q_p$-étale cohomology is a crystalline Galois representation. We then also establish a truncated version of the above results in more general setting of smooth $d$-de Rham-proper stacks over $\mathcal O_K$: here we only require first $d$ de Rham cohomology groups be finitely-generated over $\mathcal O_K$. As an application, we deduce a certain purity-type statement for étale $\mathbb Q_p$-cohomology of Raynaud generic fiber, as well as crystallinity of a first several étale cohomology groups in the presence of a Cohen--Macauley model over $\mathcal O_K$ in the schematic setting.
