Ｄｅｌｉｇｎｅ－Ｌｕｓｚｔｉｇ　多様体とＦａｒｇｕｅｓ－Ｆｏｎｔａｉｎｅ　曲線

在2020年4月至2021年3月期间，我们与我的同事Teruhisa Koshikawa一起开发了A_inf上同源的相对版本。首先介绍一些背景知识：给定整数环上的一个适当的光滑形式方案X，Bhatt-Morrow-Scholze构造了一个专门研究其他p-进上同调理论的A_inf-模的复形(他们的工作发表于2018年)。在Koshikawa和我最近的工作中，我们将这一结构推广到相对情况。简而言之，这意味着对于p-进形式方案f：X-&gt；Y的光滑态射，我们构造了一个复形(使用Decalage函子)，它住在Y的进向普通纤维的原位置上，它插值着De Rham复形。虽然我们的方法类似于Bhatt-Morrow-Scholze的方法，但在这种设置中出现了一个新的对象：Topoi的纤维化产物。这种设置的一个不同之处(与BMS相比)是，结果只有在几乎不明确的情况下才可能(由于PRO-ETALE拓扑的较高上同调群中的几乎非零元素)。我们工作的一个结果是存在一个相对Hodge-Tate谱序列，它推广了Caraiani-Scholze(DVR设置)和Abbe-Gros(方案设置)构造的谱序列。此外，我们还将我们的相对A_inf上同调与Bhatt-Scholze发展的棱柱/Q-晶体理论进行了比较。

During the period April 2020-March 2021, together with my collegue Teruhisa Koshikawa, we developed a relative version of A_Inf-cohomology. First some background: Given a proper smooth formal scheme X over the ring of integers, Bhatt-Morrow-Scholze constructed a complex of A_Inf-modules which specializes to other p-adic cohomology theories (their work published in 2018). In recent work of Koshikawa and myself we generalize this construction to the relative situation. Inshort, this means that for a smooth morphism of p-adic formal schemes f: X -> Y, we construct a complex (using the decalage functor) living on the pro-etale site of the adic generic fiber of Y, which interpolates the de Rham complex. Although, our methods are similar to that of Bhatt-Morrow-Scholze, there is the appearance of a new object in this setup: fibered product of topoi. One difference in this setup (compared to BMS) is that results are only possible up to almost ambiguity (due to almost non-zero elements in higher cohomology groups for the pro-etale topology). One consequence of our work is the existence of a relative Hodge-Tate spectral sequence which generalizes the ones constructed by Caraiani-Scholze (dvr setting) et Abbes-Gros (scheme setting). Moreover we compare our relative A_Inf-cohomology with the prismatic/q-crystalline theory developed by Bhatt-Scholze.