Bhargav Bhatt, Jacob Lurie
The goal of this paper is to study the absolute prismatic cohomology of $p$-adic formal schemes. We do so by recasting the notion of a prismatic crystal on $\mathrm{Spf}(\mathbf{Z}_p)$ in terms of quasicoherent sheaves on a geometric object we call the Cartier-Witt stack.
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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.
We introduce a logarithmic variant of the notion of $δ$-rings, which we call $δ_{\log}$-rings, and use it to define a logarithmic version of the prismatic site introduced by Bhatt and Scholze. In particular, this enables us to construct the Breuil-Kisin cohomology in the semistable case.
Prismatic cohomology of rigid analytic spaces over de Rham period ring
本文提出在Fontaine de Rham周期环上为具有l.c.i奇点的刚性解析空间定义prismatic上同调，扩展了相关上同调理论。
Overconvergent prismatic cohomology
本文定义过收敛棱柱与过收敛棱柱上同调，证明其可特化为刚性、de Rham 及 étale 等 $p$-进上同调，并给出与之关联的过收敛 $A\Omega$ 复形。
Logarithmic prismatic cohomology II
本文继续研究对数棱镜上同调，证明其导出版本的 de Rham 比较与 étale 比较，推广 Bhatt–Scholze 的结果，并构造 Nygaard 滤链，阐明 $F$-晶体与 $\mathbb{Z}_p$-局部系统的联系。
Prismatic cohomology and de Rham-Witt forms
本文为任意棱镜构造Fontaine映射的类似物，在完美情形下定义de Rham-Witt形式到棱镜上同调的典范映射并证明其为同构，进而给出$p$-完备多项式代数棱镜上同调的显式描述。
Prismatic cohomology and $A_{\inf}$-cohomology with coefficients
针对混合特征perfectoid域整数环上的光滑p-进形式概形，证明局部有限自由棱镜晶体的棱镜上同调与泛纤维proétale景上对应相对Breuil-Kisin-Fargues模的A_inf-上同调同构，借助上同调降阶用q-Higgs模给出前者整体描述，并讨论相关相容性。
We introduce the notion of a prism, which may be regarded as a "deperfection" of the notion of a perfectoid ring. Using prisms, we attach a ringed site -- the prismatic site -- to a $p$-adic formal scheme. The resulting cohomology theory specializes to (and often refines) most known integral $p$-adic cohomology theories. As applications, we prove an improved version of the almost purity theorem allowing ramification along arbitrary closed subsets (without using adic spaces), give a co-ordinate free description of $q$-de Rham cohomology as conjectured by the second author, and settle a vanishin…
We investigate the maximal finite length submodule of the Breuil-Kisin prismatic cohomology of a smooth proper formal scheme over a p-adic ring of integers. This submodule governs pathology phenomena in integral p-adic cohomology theories. Geometric applications include a control, in low degrees and mild ramifications, of (1) the discrepancy between two naturally associated Albanese varieties in characteristic p, and (2) kernel of the specialization map in p-adic étale cohomology. As an arithmetic application, we study the boundary case of the theory due to Fontaine-Laffaille, Fontaine-Messing…
We characterize the relative prismatic cohomology of Bhatt and Scholze by a universal property by endowing it with the additional structure of a ``derived $δ$-ring". This involves introducing an analogue of prismatic envelopes in the setting of filtered derived commutative rings and applying this construction to the Hodge filtration on infinitesimal cohomology. In particular, we recover relative prismatic cohomology from infinitesimal cohomology via a purely algebraic process.
Log prismatic $F$-crystals and realization functors
本文在边界除子含水平分量的情形下，构造并研究从对数棱镜F晶体范畴到其他p进上同调理论系数对象范畴的实现函子。
Frobenius height of prismatic cohomology with coefficients
本文研究棱镜 F-晶体在光滑真推前上的 Frobenius 算子，证明第 i 个推前的 Frobenius 高度最多增加 i，并利用棱镜 F-规范给出自包含证明。
Cartier smoothness in prismatic cohomology
本研究提出$p$-Cartier光滑代数概念，推广光滑代数范畴并涵盖完美胚基上赋值环，借助棱柱上同调给出该光滑性的多种刻画，同时推导了该条件下合成与平展上同调的比较定理。
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。
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。
This short note regards an observation about the recent theory of prismatic cohomology developed by Bhatt and Scholze. In particular, by applying a functor of Mandell, we see that the étale comparison theorem in the prismatic theory reproduces the $p$-adic homotopy type for a smooth proper complex variety with good reduction mod $p$.
Wild Brauer classes via prismatic cohomology
本文基于棱镜上同调与 Newton-above-Hodge 结果，证明了具有良好约化的 $p$-adic 簇上非零全局 2-形式的存在性蕴含 $p$-挠 Brauer 类的存在，并将前人工作推广至普通约化情形之外。
Prismatic cohomology relative to $δ$-rings
本文发展了相对于δ环的棱柱和syntomic上同调，证明了相对棱柱上同调实际上不依赖棱柱结构，仅依赖于底层δ环。提供了三种等价的棱柱上同调定义，并应用于滤环的棱柱上同调理论。
Prismatic Steenrod operations and arithmetic duality on Brauer groups
本文构建并分析了
Prismatic crystals over the de Rham period sheaf
本文研究了离散赋值环棱晶景上的de Rham周期层晶体，利用对数联络和Kummer塔上的Sen-Fontaine理论，通过近de Rham表示对其进行分类，并比较了其棱晶上同调与Galois上同调。
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.
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。
Finiteness and Duality for the cohomology of prismatic crystals
本文证明紧合光滑p-进形式概形上棱柱晶体的上同调为具有次数[0,2n] tor-振幅的完美复形，并利用希格斯模的局部描述建立了约化棱柱晶体的庞加莱对偶。
Prismatic crystals and $p$-adic Riemann--Hilbert correspondence
该研究系统分析光滑及半稳定形式概形的棱晶位形上的Δ₊^dR-晶体，通过分层计算与幂零/增强联络分类局部晶体，结合p进黎曼-希尔伯特函子及无限维Sen理论将结果整体化，用ℬ₊^dR-局部系统分类整体绝对晶体。
Logarithmic Prismatic Cohomology via Logarithmic THH
受Bhatt-Morrow-Scholze启发，利用对数THH滤过定义Nygaard完备化对数棱镜上同调，并将拟正则半完美体的对数THH描述为普通Hochschild同调的1参数形变。
TR and the $r$-Nygaard filtered prismatic cohomology
本文定义并研究绝对棱柱上同调上的 $r$-Nygaard 滤过，并将其用于分析 TR 的 motif 滤过及与拓扑循环同调、de Rham–Witt 复形的比较。
Essential dimension via prismatic cohomology
本文利用 Bhatt-Scholze 棱镜上同调证明复代数簇模 p 上同调在 Zariski 开集上的维数下界，并由此得到覆盖的 p-本质维数下界，进而证明阿贝尔簇模 p 同调覆盖及 Hermitian 对称域的 p-同余覆盖在 p 充分大时的 p-不可压缩性。
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。
A prismatic approach to crystalline local systems
本文用棱镜方法证明光滑 p-进形式概形上整晶体局部系统等价于棱镜 F-晶体，并由此在相对情形、一般系数及分歧基域下证明 Fontaine C_crys-猜想，同时建立棱镜 F-晶体上同调的比较定理、庞加莱对偶性与弗罗贝尼乌斯同源等基础结果。
Dieudonné theory via classifying stacks and prismatic $F$-gauges
本文将叠论方法引入迪厄多内理论，重构完美域上p幂秩有限交换群概形的经典迪厄多内模，给出相关同构的简洁新证明，引入棱镜迪厄多内F-标架完成拟合冲基上对应群概形分类并推广已有结论。
Log prismatic $F$-crystals and purity
本文研究了半稳定$p$-adic局部系统与prismatic $F$-晶的纯度性质，通过分析对数prisms推导出半稳定局部系统的纯度定理。
Absolute calculus and prismatic crystals on cyclotomic rings
本研究在Witt向量环上提出绝对微积分概念，证明有限自由模上弱幂零Δ-联络等价于分圆环上的棱柱向量丛，并得到带Frobenius结构的Δ-联络与结晶表示格的等价性，还探讨了两类棱柱晶体情形。
Height 1 Group Schemes and Prismatic F-Gauges
本文研究高度为1的有限平坦群方案的晶格F-gauge，并应用该结果描述Berthelot-Breen-Messing的晶格Dieudonné模，以及通过Hoobler型序列恢复Bragg-Olsson关于平坦上同调的结果。
Completed prismatic $F$-crystals and crystalline $\mathbf{Z}_p$-local systems
本文在光滑 $p$-进形式概形上引入完备棱镜 $F$-晶体，并证明其与晶体 étale $\mathbf{Z}_p$-局部系统范畴等价，推广了 Bhatt-Scholze 的结果。
Let $\calO_K$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ with a perfect residue field. In this paper, for a semi-stable $p$-adic formal scheme $\frakX$ over $\calO_K$ with rigid generic fibre $X$ and canonical log structure $\calM_{\frakX} = \calO_{\frakX}\cap\calO_X^{\times}$, we study Hodge--Tate crystals over the absolute logarithmic prismatic site $(\frakX,\calM_{\frakX})_{\Prism}$. As an application, we give an equivalence between the category of rational Hodge--Tate crystals on the absolute logarithmic prismatic site $(\frakX,\calM_{\frakX})_{\Prism}$ and the c…
Prismatic Crystals for schemes in characteristic $p$
构造棱镜包络层，证明棱镜晶体与可积拓扑拟幂零 $p$-联络等价，并识别棱镜上同调为 de Rham 复形；通过提升的 Ogus-Vologodsky 函子给出局部等价，进而得到 l.c.i. 概形上棱镜晶体与晶体晶体的整体等价。
Hodge--Tate prismatic crystals and Sen theory
本文研究了$\mathcal{O}_K$绝对（对数）prismatic site上的Hodge-Tate crystals，通过配备特定小自同态的模完成分类，并在非Galois Kummer塔上构造Sen theory以实现对有理Hodge-Tate crystals的分类。
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…
We prove that, modulo any power of a prime $p$, the absolute integral closure of an excellent noetherian domain is Cohen-Macaulay. A graded analog is also established, yielding variants of Kodaira vanishing "up to finite covers" in mixed characteristic. Our main tools are (log) prismatic cohomology (which yields a Frobenius action in mixed characteristic) and the $p$-adic Riemann-Hilbert functor for constructible étale $\mathbf{F}_p$-sheaves on varieties over a $p$-adic field (which almost controls perfectified prismatic cohomology).
Prismatic crystals for smooth schemes in characteristic $p$ with Frobenius lifting mod $p^2$
本文研究了特征p下光滑方案的棱柱晶体与p-连接之间的等价性，并扩展了Ogus的工作，同时讨论了Hodge-Tate丛的平凡化问题。
Thomason filtration via $T(1)$-local $\mathrm{TC}$
本文利用棱镜上同调与下降理论为任意动画交换环构造 $T(1)$-局部 $\mathrm{TC}$ 的自然滤过，并由此恢复 Thomason 谱序列与棱镜上同调的 étale 比较。
Logarithmic TC via the Infinite Root Stack and the Beilinson Fiber Square
本文利用饱和下降方法，通过无限根层统一表达多种对数上同调理论，并证明对数棱镜上同调的Nygaard完备化与对数TC一致，建立Beilinson纤维平方的对数版本。
Rational Hodge--Tate prismatic crystals of quasi-l.c.i algebras and non-abelian $p$-adic Hodge theory
本文研究了准理Hodge-塔普里扎晶体与非阿贝尔p进Hodge理论的关系，通过范畴等价性揭示了新的理论结果。
In this note, we introduce and study the Cartier--Witt stack $\mathrm{WCart}_X$ attached to a $p$-adic formal scheme $X$ as well as some variants. In particular, we reinterpret the notion of prismatic crystals on $X$ and their cohomology in terms of quasicoherent sheaf theory on $\mathrm{WCart}_X$ in favorable situations.
P-adic Simpson correpondence via prismatic crystals
本文在光滑p进形式概形上建立了有理Hodge–Tate棱镜晶体与增强希格斯丛之间的整体等价，并构造了从增强希格斯丛到广义表示的完全忠实逆Simpson函子。
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。
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。
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。
A stacky comparison of the Hodge and Nygaard filtrations
本文通过Drinfeld与Bhatt–Lurie的栈化方法，将棱镜上同调的Nygaard滤链与de Rham上同调的Hodge滤链的比较结果推广到任意规范系数情形，并发展出能同时捕捉共轭滤链与Sen算子的衍射Hodge上同调栈化理论。
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。
In this paper we investigate the arithmetic aspects of the theory of $\mathcal{E}_K^\dagger$-valued rigid cohomology introduced and studied in [11,12]. In particular we show that these cohomology groups have compatible connections and Frobenius structures, and therefore are naturally $(φ,\nabla)$-modules over $\mathcal{E}_K^\dagger$ whenever they are finite dimensional. We also introduce a category of `absolute' coefficients for the theory; the same results are true for cohomology groups with coefficients. We moreover prove a $p$-adic version of the weight monodromy conjecture for smooth (not …
Logarithmic Gysin sequences for regular immersions
该研究针对概形正则浸入与fs对数概形上同调理论，用对数紧化替代开补集构建对数Gysin序列，证明多类上同调理论均满足该序列。
Prismatization
本文构造并研究p进整数环形式谱的三种棱镜化函子，旨在为棱镜上同调提供类似F-规范的系数范畴。
Absolute Poincaré duality in étale cohomology
本文通过局部完全交态射的迹映射形式化，将étale上同调中的Poincaré对偶从光滑概形扩展至正则概形。
It is known that the Picard group of a complex manifold can be expressed as a Deligne cohomology group. One may wonder if the same holds for the Picard group of a smooth algebraic variety and Deligne-Beilinson cohomology but this is not true, as already remarked by M. Saito. We explain how one has to modify the latter, show that the Picard group can be expressed by absolute Hodge cohomology, too, and introduce an intermediate object between Picard group and usual Deligne-Beilinson cohomology group. Similarly as in the case of Deligne cohomology one can relate line bundles with a regular connec…
The primary goal of this paper is to identify syntomic complexes with the $p$-adic étale Tate twists of Geisser--Schneider--Sato on regular $p$-torsionfree schemes. Our methods apply naturally to a broader class of schemes that we call "$F$-smooth". The $F$-smoothness of regular schemes leads to new results on the absolute prismatic cohomology of regular schemes.
Crystals and Chern classes
本文研究晶体上同调及相对棱柱上同调中带晶体结构的凝聚层与完美复形的陈类，证明前者陈类消失、后者为指数受 Bernoulli 数一致约束的挠类。
We interpret syntomic cohomology of Nekovář-Nizioł as a $p$-adic absolute Hodge cohomology. This is analogous to the interpretation of Deligne-Beilinson cohomology as an absolute Hodge cohomology by Beilinson and generalizes the results of Bannai and Chiarellotto, Ciccioni, Mazzari in the good reduction case, and of Yamada in the semistable reduction case. This interpretation yields a simple construction of the syntomic descent spectral sequence and its degeneration for proper and smooth varieties. We introduce syntomic coefficients and show that in dimension zero they form a full triangulated…
Products in absolute Hodge cohomology and mixed Hodge formality
本文证明混合Hodge形式性蕴含绝对Hodge上同调的平方零扩张分解，定义类似Massey积的阻碍积，并关联乘法Chow-Künneth分解。
We extend the notion of absolute subsets of Betti moduli spaces of smooth algebraic varieties to the case of normal varieties. As a consequence we prove that twisted cohomology jump loci in rank one over a normal variety are a finite union of translated subtori. We show that the same holds for jump loci twisted by a unitary local system in the case where the underlying variety $X$ is projective with $H^1(X,\mathbb{Q})$ pure of weight one. Lastly, we study the interaction of these loci with Hodge theoretic data naturally associated to the representation variety of fundamental groups of smooth p…
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