Teruhisa Koshikawa
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.
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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.
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.
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。
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。
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。
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…
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。
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。
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).
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。
A note on prismatic sites for p-quasisyntomic rings
研究p-拟合成环R的绝对棱柱位点\(R_\Prism\)中对象覆盖最终对象的条件，引入横向对象及相对拟正则半完全oid覆盖，其棱柱上同调能产生有积的对象。
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。
Logarithmic $A_{\mathrm{inf}}$-cohomology, Part II
本文为充分对数光滑的 $p$-进形式概形建立带系数的对数 $A_{\mathrm{inf}}$-上同调理论，证明与 étale、de Rham、晶体上同调的比较同构，并由此给出半稳定局部系统 $C_{\mathrm{st}}$ 猜想的新证明。
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。
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。
$\mathbb{Z}_p$-lattices in semistable Galois representations
该研究证明对数棱柱F-晶体范畴与半稳定Galois表示中的ℤₚ-格范畴等价，并借助Breuil–Kisin模的对数版本用线性代数数据描述这类伽罗瓦表示，为数论中伽罗瓦表示研究提供新的范畴等价工具。
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 give algorithms of computing bases of logarithmic cohomology groups for square-free polynomials in two variables. (Fixed typos of v1)
In order to study $p$-adic étale cohomology of an open subvariety $U$ of a smooth proper variety $X$ over a perfect field of characteristic $p>0$, we introduce new $p$-primary torsion sheaves. It is a modification of the logarithmic de Rham-Witt sheaves of $X$ depending on effective divisors $D$ supported in $X-U$. Then we establish a perfect duality between cohomology groups of the logarithmic de Rham-Witt cohomology of $U$ and an inverse limit of those of the mentioned modified sheaves. Over a finite field, the duality can be used to study wild ramification class field theory for the open su…
Purity for flat cohomology
本文证明完全交 Noether 局部环上有限平坦交换群系数的局部平坦上同调在维数以下消失，并借助 perfectoid 环、arc descent 与棱柱 Dieudonné 理论推广 Gabber–Thomason 纯净性。
A log-motivic cohomology for semistable varieties and its $p$-adic deformation theory
本文提出了一种半稳定概形的log-动机上同调理论，并探讨了其p进变形理论，通过研究Hyodo-Kato类与Hodge滤波的关系，推广了Bloch-Esnault-Kerz的结论。
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。
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…
On logarithmic Poisson cohomology of a degenerate Poisson bivector in affine plane
本文研究了退化双矢量的古典Poisson上同调与对数Poisson上同调的同构性，通过确定对数Hamilton算子和上同调复形计算相关上同调不变量。
We characterize quasihomogeneity of isolated singularities by the injectivity of the map induced by the first differential of the logarithmic differential complex in the top local cohomology supported in the singular point.
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…
In this article we prove a semistable version of the variational Tate conjecture for divisors in crystalline cohomology, stating that a rational (logarithmic) line bundle on the special fibre of a semistable scheme over $k [\![ t ]\!]$ lifts to the total space if and only if its first Chern class does. The proof is elementary, using standard properties of the logarithmic de Rham-Witt complex. As a corollary, we deduce similar algebraicity lifting results for cohomology classes on varieties over global function fields. Finally, we give a counter example to show that the variational Tate conject…
On Logarithmic Poisson and De Rham Cohomology Groups of a Class of Inhomogeneous Divisors
本文研究仿射平面中一类非齐次自由除子的对数泊松与德拉姆上同调，利用显式基构造复形并计算上同调，证明除次数1外两者自然同构，次数1时为分裂一维扩张。
We show that classical Chern classes from higher ($p$-adic) $K$-theory to syntomic cohomology extend to logarithmic syntomic cohomology. These Chern classes are compatible -- in a suitable sense -- with addition, products, and $λ$-operations. They are also compatible with the canonical Gysin sequences and, via period maps, with logarithmic étale Chern classes. Moreover, they induce logarithmic crystalline Chern classes. This uses as a critical new ingredient the recent comparison of syntomic cohomology with $p$-adic nearby cycles and $p$-adic motivic cohomology.
In this paper, we develop the theory of relative log convergent cohomology. We prove the coherence of relative log convergent cohomology in certain case by using the comparison theorem between relative log convergent cohomlogy and relative log crystalline cohomology, and we relates relative log convergent cohomology to relative rigid cohomology to show the validity of Berthelot's conjecture on the coherence and the overconvergence of relative rigid cohomology for proper smooth families when they admit nice proper log smooth compactification to which the coefficient extends logarithmically.
Let Y be a divisor on a smooth algebraic variety X. We investigate the geometry of the Jacobian scheme of Y, homological invariants derived from logarithmic differential forms along Y, and their relationship with the property that Y is a free divisor. We consider arrangements of hyperplanes as a source of examples and counterexamples. In particular, we make a complete calculation of the local cohomology of logarithmic forms of generic hyperplane arrangements.
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.
Construction of logarithmic cohomology theories I
本文借助 $\mathbb{F}_1$ 上的对数动机同伦论构造严格正规交叉边界对数概形的上同调理论，恢复开补集 K-理论与对数 THH，并证明非正则概形 K-理论可表且引入对数分圆迹。
In this article, we analyze the connection between the Log De Rham Cohomology of an fs (not necessary log smooth) log scheme $Y$ over $\mathbb C$ (for $Y$ admitting an exact closed immersion into an fs log smooth log scheme over $\mathbb C$), its Log Infinitesimal Cohomology $H^{^.}(Y^{log}_{inf}, \mathcal O_{Y^{log}_{inf}})$, and its Log Betti Cohomology, which is the Cohomology of its associated Kato-Nakayama topological space $Y^{an}_{log}$, and we prove that they are isomorphic. These results are the log scheme analogues of two classical comparison theorems.
We show that the direct image of the filtered logarithmic de Rham complex is a direct sum of filtered logarithmic complexes with coefficients in variations of Hodge structures, using a generalization of the decomposition theorem of Beilinson, Bernstein and Deligne to the case of filtered $D$-modules. The advantage of using the logarithmic complexes is that we have the strictness of the Hodge filtration by Deligne after taking the cohomology group in the projective case. As a corollary, we get the total infinitesimal invariant of a (higher) cycle in a direct sum of the cohomology of filtered lo…
Construction of logarithmic cohomology theories II: On Chow groups
本文通过展示 toric 变形的 Chow 群技术结果，为第一部分提供关键支持。
We construct correspondences in logarithmic Hodge theory over a perfect field of arbitrary characteristic. These are represented by classes in the cohomology of sheaves of differential forms with log poles and, notably, log zeroes on cartesian products of varieties. From one perspective this generalizes work of Chatzistamatiou and Rülling, who developed (non-logarithmic) Hodge correspondences over perfect fields of arbitrary characteristic; from another we provide partial generalizations of more recent work of Binda, Park and Østvær on logarithmic Hodge correspondences by relaxing finiteness a…
Let $K$ be a local field, $X$ the Drinfel'd symmetric space $X$ of dimension $d$ over $K$ and ${\mathfrak X}$ the natural formal ${\mathcal O}_K$-scheme underlying $X$; thus $G={\rm GL}\sb {d+1}(K)$ acts on $X$ and ${\mathfrak X}$. Given a $K$-rational $G$-representation $M$ we construct a $G$-equivariant subsheaf ${\mathcal M}^0_{{\mathcal O}_{\dot{K}}}$ of ${\mathcal O}_K$-lattices in the constant sheaf $M$ on ${\mathfrak X}$. We study the cohomology of sheaves of logarithmic differential forms on $X$ (or ${\mathfrak X}$) with coefficients in ${\mathcal M}^0_{{\mathcal O}_{\dot{K}}}$. In the…
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