Zijian Yao
We provide a simple approach for the crystalline comparison of Ainf-cohomology, and reprove the comparison between crystalline and p-adic etale cohomology for formal schemes in the case of good reduction.
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The goal of this short paper is to give a slightly different perspective on the comparison between crystalline cohomology and de Rham cohomology. Most notably, we reprove Berthelot's comparison result without using pd-stratifications, linearisations, and pd-differential operators.
A remark on crystalline cohomology
本文基于光滑代数可唯一提升至相干同伦的观察，提出研究晶体上同调的新途径。
Crystalline cohomology of rigid analytic spaces
本文针对未必光滑的刚性解析空间，引入系数在p进域或Fontaine的de Rham周期环中的无穷小上同调，推广了晶体上同调理论。
In this article, we prove the comparison theorem between the relative log de Rham-Witt cohomology and the relative log crystalline cohomology for a log smooth saturated morphism of fs log schemes satisfying certain condition. Our result covers the case where the base fs log scheme is etale locally log smooth over a scheme with trivial log structure or the case where the base fs log scheme is hollow, and so it generalizes the previously known results of Matsuue. In Appendix, we prove that our relative log de Rham-Witt complex and our comparison map are compatible with those of Hyodo-Kato.
Motivated by applications in point counting algorithms using p-adic cohomology, we give an explicit description of integral lattices in rigid cohomology spaces that p-adically approximate logarithmic crystalline cohomology modules. These lattices are expressed in terms of the global sections of the twisted logarithmic de Rham complex. We prove the main theorem for smooth proper hypersurfaces with a smooth hyperplane section, then deduce the result for the quotient of such a pair in weighted projective space. We show how these results may be used to reduce the necessary p-adic precision with wh…
This note discusses some examples showing that the crystalline cohomology of even very mildly singular projective varieties tends to be quite large. In particular, any singular projective variety with at worst ordinary double points has infinitely generated crystalline cohomology in at least two cohomological degrees. These calculations rely critically on comparisons between crystalline and derived de Rham cohomology.
For a perfect field $k$ of characteristic $p>0$ and a smooth and proper formal scheme $\mathscr{X}$ over the ring of integers of a finite and totally ramified extension $K$ of $W(k)[1/p]$, we propose a cohomological construction of the Breuil-Kisin modules attached to the $p$-adic étale cohomology $H^i_{\mathrm{ét}}(\mathscr{X}_{\overline{K}},\mathbf{Z}_p)$. We then prove that our proposal works when $p>2$, $i < p-1$, and the crystalline cohomology of the special fiber of $\mathscr{X}$ is torsion-free in degrees $i$ and $i+1$.
Let $K$ be a field of characteristic zero complete for a discrete valuation, with perfect residue field of characteristic $p>0$, and let $K^+$ be the valuation ring of $K$. We relate the log-crystalline cohomology of the special fibre of certain affine $K^+$-schemes $X=\text{Spec}(R)$ with semi-stable reduction to the Galois cohomology of the fundamental group of the geometric generic fibre $π_1(X_{\bar{K}})$ with coefficients in a Fontaine ring constructed from $R$. This is based on Faltings' theory of almost étale extensions.
Computing Crystalline Cohomology and p-Divisible Groups for Curves over Finite Fields
本文提出了一种计算有限域上曲线的p-可除群的算法，基于晶体上同调的Frobenius和Verschiebung算子，改进了Tuitman的p进点计数算法。
For a strictly semistable log scheme $Y$ over a perfect field $k$ of characteristic $p$ we investigate the canonical Čech spectral sequence $(C)_T$ abutting to the Hyodo-Kato (log crystalline) cohomology $H_{crys}^*(Y/T)_{\mathbb{Q}}$ of $Y$ and beginning with the log convergent cohomology of its various component intersections $Y^i$. We compare the filtration on $H_{crys}^*(Y/T)_{\mathbb{Q}}$ arising from $(C)_T$ with the monodromy operator $N$ on $H_{crys}^*(Y/T)_{\mathbb{Q}}$. We also express $N$ through residue maps and study relations with singular cohomology. If $Y$ lifts to a proper str…
Injectivity failure in crystalline comparisons
本文针对正特征光滑仿射簇，识别刚性上同调到有理化晶体上同调比较态射单射性的斜率障碍，否定Esnault-Kisin-Petrov相关问题，同时建立特定子空间和1度上同调单射性，还获分离结果、新整比较定理及分数p-进Tate扭的定义与计算。
An arithmetic étale-crystalline comparison with coefficients in crystalline local systems
本文利用栈化方法将 Colmez--Niziol 的算术平展-晶体比较定理推广到任意晶体局部系系数，并建立带系数的 Beilinson 纤维方块及 syntomic 上同调与 p 进平展上同调的比较，同时描述 Z_p 上完美 F-规范的等ogeny范畴。
We prove the finiteness of crystalline cohomology of higher level. An important ingredient is a "higher de Rham complex" and a kind of Poincaré lemma for it.
An integral comparison of crystalline and de Rham cohomology
本文借助 Drinfeld 与 Bhatt–Lurie 的棱柱上同调叠视角，建立了带棱柱 $F$-晶体完美复形系数的晶体与 de Rham 上同调整性比较。
Delta characters and crystalline cohomology of abelian schemes
本文研究了阿贝尔方案上晶体上同调的最小滤子子isocrystal，通过算术喷泉空间和delta特征理论，证明了Borger和Saha构造的delta isocrystal与晶体上同调的最小子isocrystal同构。
Ramification bounds via Wach modules and q-crystalline cohomology
本文通过Wach模建立模p Galois表示的分歧界，并利用q-晶体上同调改进光滑真p-adic形式概形的高阶étale上同调群的分歧估计。
Dieudonné Theory via Cohomology of Classifying Stacks
本文证明有限平坦群概形分类栈的第二次晶体上同调恢复其Dieudonné模，并计算阿贝尔簇及$p$可除群情形，推广至混合特征。
A log prismatic-crystalline comparison theorem
该研究通过比较log prismatic和log crystalline上同调，证明了半稳定情况下$C_{\mathrm{st}}$猜想的成立。
This paper studies the derived de Rham cohomology of F_p and p-adic schemes, and is inspired by Beilinson's recent work. Generalising work of Illusie, we construct a natural isomorphism between derived de Rham cohomology and crystalline cohomology for lci maps of such schemes, as well logarithmic variants. These comparisons give derived de Rham descriptions of the usual period rings and related maps in p-adic Hodge theory. Placing these ideas in the skeleton of Beilinson's construction leads to a new proof of Fontaine's crystalline conjecture and Fontaine-Jannsen's semistable conjecture.
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.
Given a proper, smooth (formal) scheme over the ring of integers of $\mathbb C_p$, we prove that if the crystalline cohomology of its special fibre is torsion-free then the $p$-adic étale cohomology of its generic fibre is also torsion-free. In this announcement we sketch the proof, which relies on the construction of a new cohomology theory interpolating between crystalline and étale cohomology. Further details and results, including a comparison isomorphism, will be presented in the full forthcoming article.
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 proper scheme or formal scheme over an unramified, complete DVR of mixed characteristics we prove a comparison isomorphism relating etale cohomology of the generic fiber with values in a crystalline etale sheaf to the crystalline cohomology of its special fiber with values in the associated F-isocrystal.
Etale and crystalline companions, I
本文研究了在有限域上光滑概形的晶体质与étale上同调之间的伴侣关系，证明了晶体质系数对象具有étale伴侣，并探讨了相关辅助结果。
We establish a comparison isomorphism between prismatic cohomology and derived de Rham cohomology respecting various structures, such as their Frobenius actions and filtrations. As an application, when $X$ is a proper smooth formal scheme over $\mathcal O_K$ with $K$ being a $p$-adic field, we improve Breuil--Caruso's theory on comparison between torsion crystalline cohomology and torsion étale cohomology.
Crystalline condition for $A_{\mathrm{inf}}$-cohomology and ramification bounds
本文针对 $p$-进形式概形，提出系列条件 ($\mathrm{Cr}_s$) 控制棱镜上同调上的伽罗瓦作用，证明晶体性并推广分歧界。
We prove that rigid cohomology can be computed as the cohomology of a site analogous to the crystalline site. Berthelot designed rigid cohomology as a common generalization of crystalline and Monsky-Washnitzer cohomology. Unfortunately, unlike the former, the functoriality of the theory is not built-in. We defined somewhere else the "overconvergent site" which is functorially attached to an algebraic variety and proved that the category of modules of finite presentation on this ringed site is equivalent to the category of over- convergent isocrystals on the variety. We show here that their coh…
Let $X\to Y^0$ be an abelian prime-to-$p$ Galois covering of smooth schemes over a perfect field $k$ of characteristic $p>0$. Let $Y$ be a smooth compactification of $Y^0$ such that $Y-Y^0$ is a normal crossings divisor on $Y$. We describe a logarithmic $F$-crystal on $Y$ whose rational crystalline cohomology is the rigid cohomology of $X$, in particular provides a natural $W[F]$-lattice inside the latter; here $W$ is the Witt vector ring of $k$. If a finite group $G$ acts compatibly on $X$, $Y^0$ and $Y$ then our construction is $G$-equivariant. As an example we apply it to Deligne-Lusztig va…
Logarithmic Crystalline Representations
本文提出对数Fontaine-Faltings模块和对数晶体质谱的构造方法，扩展了Faltings在1989年关于étale上同调与晶体质谱比较定理的研究。
Divided Powers and Derived De Rham Cohomology
本研究构建导出除幂代数形式体系，在该框架下重新研究导出德拉姆与晶体上同调，给出其泛性质刻画，验证其与经典理论的一致性，并推导相关代数几何工具的应用。
In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of $\ell$-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In this paper, we confirm Ogus' prediction for some families of abelian varieties. These families include abelian varieties that have both prime dimension and nontrivial endomorphism ring. The proof is based on …
Revisiting derived crystalline cohomology
本文引入 animated PD-对与非完备 animated 棱镜理论，建立导出晶体上同调的多种构造及比较定理，并将相关比较推广到 quasisyntomic 概形。
We survey crystalline cohomology, crystals, and formal group laws with an emphasis on geometry. We apply these concepts to K3 surfaces, and especially to supersingular K3 surfaces. In particular, we discuss stratifications of the moduli space of polarized K3 surfaces in positive characteristic, Ogus' crystalline Torelli theorem for supersingular K3 surfaces, the Tate conjecture, and the unirationality of K3 surfaces.
The crystalline cohomology of covers with a cyclic $p$-Sylow subgroup
将 Chevalley-Weil 关于覆盖的 de Rham 上同调的 G-结构结果推广到晶体上同调，并显式描述 G=Z/p^n 时的结构，主要工具是 Yakovlev 图论。
Given a smooth, proper family of varieties in characteristic $p>0$, and a cycle $z$ on a fibre of the family, we formulate a Variational Tate Conjecture characterising, in terms of the crystalline cycle class of $z$, whether $z$ extends cohomologically to the entire family. This is a characteristic $p$ analogue of Grothendieck's Variational Hodge Conjecture. We prove the conjecture for divisors, and an infinitesimal variant of the conjecture for cycles of higher codimension. This can be used to reduce the $\ell$-adic Tate conjecture for divisors over finite fields to the case of surfaces.
Let k be an algebraically closed field of characteristic p>0. Let W(k) be the ring of Witt vectors with coefficients in k. We prove a motivic conjecture of Milne that relates, in the case of abelian schemes, the étale cohomology with $\dbZ_p$ coefficients to the crystalline cohomology with integral coefficients, in the more general context of p-divisible groups endowed with {\it arbitrary} families of crystalline tensors over a finite, discrete valuation ring extension of W(k). This extends a result of Faltings in [Fa2]. As a main new tool we construct global deformations of p-divisible groups…
We construct Λ-adic de Rham and crystalline analogues of Hida's ordinary Λ-adic etale cohomology, and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of \Q_p, we prove appropriate finiteness and control theorems in each case. We then employ integral p-adic Hodge theory to prove Λ-adic comparison isomorphisms between our cohomologies and Hida's etale cohomology. As applications of our work, we provide a "cohomological" construction of the family of (ϕ,Γ)-modules attached to Hida's ordinary Λ-adic etale cohomology by Dee, and we give a new and purely…
A stacky approach to $p$-adic Hodge theory
本文利用叠方法推广$p$-进霍奇理论中的比较定理，建立带系数的多种上同调比较结果，发展衍射霍奇上同调的叠方法，为波恩大学硕士论文成果。
Delta Characters and Crystalline Cohomology
该文发展m移位π-典型Witt向量理论并证明其delta结构性质，进而研究阿贝尔概形对应的典范滤过等晶的秩、弱可容许性及与晶体上同调的关系，加强了Buium的相关结果。
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, b…
For a proper, smooth scheme $X$ over a $p$-adic field $K$, we show that any proper, flat, semistable $\mathcal{O}_K$-model $\mathcal{X}$ of $X$ whose logarithmic de Rham cohomology is torsion free determines the same $\mathcal{O}_K$-lattice inside $H^i_{dR}(X/K)$ and, moreover, that this lattice is functorial in $X$. For this, we extend the results of Bhatt--Morrow--Scholze on the construction and the analysis of an $A_{inf}$-valued cohomology theory of $p$-adic formal, proper, smooth $\mathcal{O}_{\overline{K}}$-schemes $\mathfrak{X}$ to the semistable case. The relation of the $A_{inf}$-coho…
We prove that, for adic étale motives over $\mathbb{C}_p$, the vector bundles on the Fargues-Fontaine curve arising from their Hyodo-Kato cohomology coincide with their de Rham-Fargues-Fontaine cohomologies, where the latter provides an overconvergent refinement of crystalline vector bundles, albeit constructed on the generic fiber. This equivalence is established in the setting of symmetric monoidal $\infty$-categories and respects the full motivic structure. Furthermore, we enrich both realizations with Galois actions, yielding $G_{\breve{\mathbb{Q}}_{p}}$-equivariant solid quasi-coherent sh…
Prismatic crystals and $q$-Higgs fields
本文类比晶体上同调，在$q$-晶体棱镜上给出棱镜晶体及其上同调的局部$q$-Higgs模描述，并讨论标量扩张、张量积及Zariski下降下的全局行为。
Logarithmic A$_{\rm inf}$-cohomology
本文将 Bhatt-Morrow-Scholze 的 A$_{\ m inf}$-上同调推广到对数完美oid基上的对数 $p$-进形式概形，证明其与多种 $p$-进上同调理论的比较定理，并引入伪饱和映射这一新概念。
On the cohomology of $p$-adic analytic spaces, I: The basic comparison theorem
本文对$p$-进域代数闭包上的光滑刚性解析簇和dagger簇证明了基本$p$-进比较定理，将稳定范围内的pro-étale上同调表为de Rham上同调的滤过Frobenius特征空间，并通过完美oid空间范畴上的层实现了几何化。
Some applications of the Nygaard filtration and quasisyntomic descent in positive characteristic
本文阐述了正特征下的quasisyntomic下降与Nygaard滤过，提出了Illusie比较定理的新方法，重新审视了Ogus比较定理，并确定了乘以$n$在Abel簇fppf上同调上的作用。
On endomorphisms of the de Rham cohomology functor
本文计算 de Rham 与晶体上同调函子的自同态模空间，并由此推导出 Drinfeld 对 Deligne--Illusie 分解的改进及进一步的函子性结论。
In this article, we suggest a categorification procedure in order to capture an analogy between Crystalline Grothendieck-Lefschetz trace formula and the cyclotomic trace map $K\rightarrow TC$ from the algebraic $K$-theory to the topological cyclic homology $TC$. First, we categorify the category of schemes to the $(2, \infty)$-category of noncommuatative schemes a la Kontsevich. This gives a categorification of the set of rational points of a scheme. Then, we categorify the Crystalline Grothendieck-Lefschetz trace formula and find an analogue to the Crystalline cohomology in the setting of non…
We construct the crystalline comparison isomorphisms for proper smooth formal schemes over an absolutely unramified base. Such isomorphisms hold for étale cohomology with nontrivial coefficients, as well as in the relative setting, i.e. for proper smooth morphisms of smooth formal schemes. The proof is formulated in terms of the pro-étale topos introduced by Scholze, and uses his primitive comparison theorem for the structure sheaf on the pro-étale site. Moreover, we need to prove the Poincaré lemma for crystalline period sheaves, for which we adapt the idea of Andreatta and Iovita. Another in…
We compare flat cohomology with crystalline syntomic complexes in two cases: 1) $p$-divisible groups over a separated $\mathbb F_p$-scheme with local finite $p$-bases, 2) semi-abelian schemes over a separated irreducible smooth curve.
Logarithmic prismatic cohomology, motivic sheaves, and comparison theorems
论文研究了对数棱柱和对数syntomic上同调在对数动机范畴中的可代表性，并探讨了其应用及比较定理。
A prismatic-etale comparison theorem in the semistable case
本文证明了半稳定 $p$-进形式概型上解析对数棱镜 $F$-晶体的 Breuil--Kisin 上同调与对应平展 $\mathbb{Z}_p$-局部系统平展上同调的比较定理，将已有晶体情形的棱镜-平展比较推广至半稳定情形。
Higher Residue Pairing for $p$-adic Isocrystals and the $p$-adic Riemann--Hilbert Correspondence
本文提出一种$p$-adic类比的higher residue pairing，通过扭曲的相对de Rham-Witt复形构建，连接晶体几何与残差理论，并为$p$-adic原始形式等理论提供基础。
Crystalline prisms: Reflections and diffractions, present and past
本文研究了棱柱位形上的晶体与 $p$-连接，提出了一种几何构造方法，用于描述棱柱 Sen 运算符，并给出了对 $p$-de Rham 复合体的 $α$-变换作用的清晰描述。
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