设 X 为光滑 p-进形式概形。我们证明，X 的一般纤维上的整晶体局部系统等价于 X 的棱镜景的解析轨迹上的棱镜 F-晶体。作为应用，我们在相对情形下，对一般系数，并允许分歧基域，给出了 Fontaine C_crys-猜想的棱镜证明。在此过程中，我们还建立了棱镜 F-晶体上同调的各种基础性结果，包括各种比较定理、庞加莱对偶性和弗罗贝尼乌斯同源。
英文摘要：
Let X be a smooth p-adic formal scheme. We show that integral crystalline local systems on the generic fiber of X are equivalent to prismatic F-crystals over the analytic locus of the prismatic site of X. As an application, we give a prismatic proof of Fontaine's C_crys-conjecture, for general coefficients, in the relative setting, and allowing ramified base fields. Along the way, we also establish various foundational results for the cohomology of prismatic F-crystals, including various comparison theorems, Poincaré duality, and Frobenius isogeny.
In this article, we define the $m$-prismatic site and the $m$-$q$-crystalline site, which are higher level analogs of the prismatic site and the $q$-crystalline site respectively. We prove a certain equivalence between the category of crystals on the $m$-prismatic site (resp. the $m$-$q$-crystalline site) and that on the prismatic site (resp. the $q$-crystalline site), which can be regarded as the prismatic (resp. the $q$-crystalline) analog of the Frobenius descent due to Berthelot and the Cartier transform due to Ogus-Vologodsky, Oyama and Xu. We also prove the equivalence between the catego…
An integral analogue of Fontaine's crystalline functor
Graduate School of Mathematical Sciences, The University of Tokyo（东京大学大学院数学科学综合研究科）；Institut des Hautes Études Scientifiques（高等科学研究所）；Department of Mathematics, Bahen Centre, University of Toronto（多伦多大学巴亨中心数学系）
Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心）；Purdue University（普渡大学）；Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）
In this paper, we prove that for any $p$-adic smooth separated formal scheme $\mathfrak X$, the category of prismatic $F$-crystals with $I$ inverted is equivalent to the category of étale $\mathbb Z_p$-local systems on the generic fiber of $\mathfrak X$. We also compare the cohomology of the corresponding coefficients.
This work is devoted to the study of integral $p$-adic Hodge theory in the context of Artin stacks. For a Hodge-proper stack, using the formalism of prismatic cohomology, we establish a version of $p$-adic Hodge theory with the étale cohomology of the Raynaud generic fiber as an input. In particular, we show that the corresponding Galois representation is crystalline and that the associated Breuil-Kisin module is given by the prismatic cohomology. An interesting new feature of the stacky setting is that the natural map between étale cohomology of the algebraic and the Raynaud generic fibers is…
The crystalline period map is a tool for linearizing $p$-divisible groups. It has been applied to study the Langlands correspondences, and has possible applications to the homotopy groups of spheres. The original construction of the period map is inherently local. We present an alternative construction, giving a map on the entire moduli stack of $p$-divisbile groups, up to isogeny, which specializes to the original local construction.
Let G be a connected split reductive group over a complete discrete valuation ring of mixed characteristic. We use the theory of intermediate extensions due to Abe-Caro and arithmetic Beilinson-Bernstein localization to classify irreducible modules over the crystalline distribution algebra of G in terms of overconvergent isocrystals on locally closed subspaces in the (formal) flag variety of G. We treat the case of SL(2) as an example.
Let $K$ be an unramified $p$-adic local field and let $W$ be the ring of integers of $K$. Let $(X,S)/W$ be a smooth proper scheme together with a normal crossings divisor. We show that there are only finitely many log crystalline $\mathbb Z_{p^f}$-local systems over $X_K\setminus S_K$ of given rank and with geometrically absolutely irreducible residual representation, up to twisting by a character. The proof uses $p$-adic nonabelian Hodge theory and a finiteness result due Abe/Lafforgue.
Let $K$ be an unramified $p$-adic local field and let $W$ be the ring of integers of $K$. Let $(X,S)/W$ be a smooth proper scheme together with a simple normal crossings divisor and fix positive integers $r$ and $f$. We show that the set of absolutely irreducible representations $π_1(X_{\bar K})\rightarrow \mathrm{GL}_r(\mathbb{Z}_{p^f})$ that come from log crystalline $\mathbb Z_{p^f}$-local systems over $(X_K,S_K)$ of rank $r$ is finite. The proof uses $p$-adic nonabelian Hodge theory and a finiteness result due Abe/Lafforgue.
Le Stum and Quirós proved the formal Poincaré lemma in crystalline cohomology of higher level using the jet complex, and applied it to give a de Rham interpretation of this cohomology. In this article, we prove the logarithmic version of the formal Poincaré lemma modulo p. Provided that each term of the log. jet complex is locally free, it gives the logarithmic version of the de Rham interpretation of the crystalline cohomology of higher level.
For a smooth and proper scheme over an artinian local ring with ordinary reduction over the perfect residue field we prove - under some general assumptions - that the relative de Rham-Witt spectral sequence degenerates and the relative crystalline cohomology, equipped with its display structure arising from the Nygaard complexes, has a Hodge-Witt decomposition into a direct sum of (suitably Tate-Twisted) multiplicative displays. As examples our main results include the cases of abelian schemes, complete intersections, varieties of K3 type and some Calabi-Yau $n$-folds.
