The de Rham-Witt and Z_p-cohomologies of an algebraic variety

Pseudovaluations on the De Rham-Witt complex

For a polynomial ring over a commutative ring of positive characteristic, we define on the associated de Rham-Witt complex a set of functions, and show that they are pseudovaluations in the sense of Davis, Langer and Zink. To achieve it, we explicitly compute products of basic elements on the complex. We also prove that the overconvergent de Rham-Witt complex can be recovered using these pseudovaluations.

过收敛德雷-维特复形的局部结构

Local structure of the overconvergent de Rham-Witt complex

研究过收敛德雷-维特复形在多项式环上的结构，分析Lazard映射的过收敛性，并在完美正特征域上的光滑代数簇局部结构。

On comparison between relative log de Rham-Witt cohomology and relative log crystalline cohomology

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.

关于分类堆的de Rham-Witt上同调

On de Rham--Witt Cohomology of Classifying Stacks

研究在特征p>0的完美域上构造一个四维正则光滑堆，其总度3的Hodge-Witt数不对称，展示了该堆的de Rham-Witt上同调群的计算与近似方法。

Duality for relative logarithmic de Rham-Witt sheaves and wildly ramified class field theory over finite fields

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…

Higher displays arising from filtered de Rham-Witt complexes

For a smooth projective scheme $X$ over a ring $R$ on which $p$ is nilpotent that meets some general assumptions we prove that the crystalline cohomology is equipped with the structure of a higher display which is a relative version of Fontaine's strongly divisible lattices. Frobenius-divisibilty is induced by the Nygaard filtration on the relative de Rham-Witt complex. For a nilpotent PD-thickening $S/R$ we also consider the associated relative display and can describe it explicitly by a relative version of the Nygaard filtration on the de Rham-Witt complex associated to a lifting of $X$ over…

A De Rham-Witt approach to crystalline rational homotopy theory

We construct the crystalline fundamental group of a semi-stable variety over a field of positive characteristic using the log De Rham-Witt complex and Navarro-Aznar's derived Thom-Whitney functor. This approach gives a relatively direct comparison theorem with the De Rham fundamental group in the liftable case. Also, we get a natural weight filtration which makes the coordinate ring of the crystalline fundamental group into a mixed isocrystal when the variety is defined over a finite field. Finally, the monodromy operator on the crystalline fundamental group is used to prove a crystalline anal…

Overconvergent de Rham-Witt cohomology for semistable varieties

We define an overconvergent version of the Hyodo-Kato complex for semistable varieties $Y$ over perfect fields of positive characteristic, and prove that its hypercohomology tensored with $\mathbb{Q}$ recovers the log-rigid cohomology when $Y$ is quasi-projective. We then describe the monodromy operator using the overconvergent Hyodo-Kato complex. Finally, we show that overconvergent Hyodo-Kato cohomology agrees with log-crystalline cohomology in the projective semistable case.

Duality for relative logarithmic de Rham-Witt sheaves on semistable schemes over $\mathbb{F}_q[[t]]$

We study duality theorems for the relative logarithmic de Rham-Witt sheaves on semi-stable schemes $X$ over a local ring $\mathbb{F}_q[[t]]$, where $\mathbb{F}_q$ is a finite field. As an application, we obtain a new filtration on the maximal abelian quotient $π^{\text{ab}}_1(U)$ of the étale fundamental groups $π_1(U)$ of an open subscheme $U \subseteq X$, which gives a measure of ramification along a divisor $D$ with normal crossing and $\text{Supp}(D) \subseteq X-U$. This filtration coincides with the Brylinski-Kato-Matsuda filtration in the relative dimension zero case.

棱镜上同调与de Rham-Witt形式

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。

Integral Monsky-Washnitzer cohomology and the overconvergent de Rham-Witt complex

In their paper which introduced Monsky-Washnitzer cohomology, Monsky and Washnitzer described conditions under which the definition can be adapted to give integral cohomology groups. It seems to be well-known among experts that their construction always gives well-defined integral cohomology groups, but this fact also does not appear to be explicitly written down anywhere. In this paper, we prove that the integral Monsky-Washnitzer cohomology groups are well-defined, for any nonsingular affine variety over a perfect field of characteristic p. We then compare these cohomology groups with overco…

Big de Rham-Witt cohomology: basic results

Let $X$ be a smooth projective $R$-scheme, where $R$ is a smooth $\Z$-algebra. As constructed by Hesselholt, we have the absolute big de Rham-Witt complex $\WΩ^*_X$ of $X$ at our disposal. There is also a relative version $\WΩ^*_{X/R}$ with $\W(R)$-linear differential. In this paper we study the hypercohomology of the relative (big) de Rham-Witt complex after truncation with finite truncation sets $S$. We show that it is a projective $\W_S(R)$-module, provided that the de Rham cohomology is a flat $R$-module. In addition, we establish a Poincaré duality theorem.

de Rham-Witt 复形的动机构造

A motivic construction of the de Rham-Witt complex

本文推广了互反层理论至 $\mathbb{Q}$-除子情形，并由此给出了 de Rham-Witt 复形的动机构造，类比于 Milnor $K$-理论的动机构造。

Integral Comparison of Monsky-Washnitzer and overconvergent de Rham-Witt cohomology

The goal of this small note is to extend a result by Christopher Davis and David Zureick-Brown on the comparison between integral Monsky-Washnitzer cohomology and overconvergent de~Rham-Witt cohomology for a smooth variety over a perfect field of positive characteristic $p$ to all cohomological degrees independent of the dimension of the base or the prime number $p$. Le but de ce travail est de prolonger un résultat de Christopher Davis et David Zureick-Brown concernant la comparaison entre la cohomologie de Monsky-Washnitzer entière et la cohomologie de de~Rham-Witt surconvergente d'une varié…

微分消逝的饱和de Rham-Witt复形

Dimensional vanishing of the saturated de Rham-Witt complex

本文研究了饱和de Rham-Witt复形在奇异方案上的维度消逝性质，证明其在存在奇点时仍保持良好的消逝性质，优于传统上同调理论。

大德拉姆-维它形式与非减方案的动机

The big de Rham-Witt forms over fields and motives of non-reduced schemes

本文通过代数循环证明了大德拉姆-维它形式群与Artin局部代数的相对改进米尔诺K群同构，并解释了其在非减方案动机上同调中的应用。

带单位根系数的饱和 de Rham-Witt 复形

Saturated de Rham-Witt complexes with unit-root coefficients

本文通过泛性质将 Bhatt-Lurie-Mathew 的饱和 de Rham-Witt 复形推广到单位根 F-晶体系数情形，并证明其存在性、拟凝聚性及与经典对象和晶体上同调的比较结果。

The big de Rham-Witt complex

This paper gives a new and direct construction of the multi-prime big de Rham-Witt complex which is defined for every commutative and unital ring; the original construction by the author and Madsen relied on the adjoint functor theorem and accordingly was very indirect. (The construction given here also corrects the 2-torsion which was not quite correct in the original version.) The new construction is based on the theory of modules and derivations over a lambda-ring which is developed first. The main result in this first part of the paper is that the universal derivation of a lambda-ring is g…

Logarithmic de Rham--Witt complexes via the Décalage operator

We provide a new formalism of de Rham--Witt complexes in the logarithmic setting. This construction generalizes a result of Bhatt--Lurie--Mathew, and agrees with those of Hyodo--Kato and Matsuue for log-smooth schemes of log-Cartier type. We then apply our formalism to obtain a more direct proof of the log crystalline comparison of A_inf-cohomology in the case of semistable reduction, which is established by Cesnavicius--Koshiwara.

等晶体与 de Rham-Witt 联络

Isocrystals and de Rham-Witt connections

本文在拓扑空间的微分分次代数层上引入可积联络概念，并在形式概形及正特征光滑概形的 de Rham-Witt 复形等有限局部投射情形下加以刻画，从而为带 Frobenius 结构的收敛与过收敛等晶体提供了新描述。

Rigid Cohomology and de Rham-Witt complexes

Let $k$ be a perfect field of characteristic $p > 0$, $W_n = W_n(k)$. For separated $k$-schemes of finite type, we explain how rigid cohomology with compact supports can be computed as the cohomology of certain de Rham-Witt complexes with coefficients. This result generalizes the classical comparison theorem of Bloch-Illusie for proper and smooth schemes. In the proof, the key step is an extension of the Bloch-Illusie theorem to the case of cohomologies relative to $W_n$ with coefficients in a crystal that is only supposed to be flat over $W_n$.

de Rham-Witt sheaves via algebraic cycles

We show that the additive higher Chow groups of regular schemes over a field induce a Zariski sheaf of pro-differential graded algebras, whose Milnor range is isomorphic to the Zariski sheaf of big de Rham-Witt complexes. This provides an explicit cycle-theoretic description of the big de Rham-Witt sheaves. Several applications are derived.

Revisiting the de Rham-Witt complex

The goal of this paper is to offer a new construction of the de Rham-Witt complex of smooth varieties over perfect fields of characteristic $p>0$. We introduce a category of cochain complexes equipped with an endomorphism $F$ of underlying graded abelian groups satisfying $dF = pFd$, whose homological algebra we study in detail. To any such object satisfying an abstract analog of the Cartier isomorphism, an elementary homological process associates a generalization of the de Rham-Witt construction. Abstractly, the homological algebra can be viewed as a calculation of the fixed points of the Be…

Additive Chow groups with higher modulus and the generalized de Rham-Witt complex

Bloch and Esnault defined additive higher Chow groups with modulus (m+1) on the level of zero cycles over a field k, denoted by TH^n(k,n;m), n,m >0. They prove that TH^n(k,n;1) is isomorphic to the group of absolute Kaehler differentials of degree n-1 over k. In this paper we generalize their result and show that TH^n(k,n;m) is isomorphic to W_mΩ^{n-1}_k, the group of degree n-1 elements in the generalized de Rham-Witt of length m over k. This complex was defined by Hesselholt and Madsen and generalizes the p-typical de Rham-Witt complex of Bloch-Deligne-Illusie. Before we can prove this theor…

Witt Vector Rings and the Relative de Rham Witt Complex

In this paper we develop a novel approach to Witt vector rings and to the (relative) de Rham Witt complex. We do this in the generality of arbitrary commutative algebras and arbitrary truncation sets. In our construction of Witt vector rings the ring structure is obvious and there is no need for universal polynomials. Moreover a natural generalization of the construction easily leads to the relative de Rham Witt complex. Our approach is based on the use of free or at least torsion free presentations of a given commutative ring $R$ and it is an important fact that the resulting objects are inde…

半稳定形式概型上 A_inf-上同调与 de Rham-Witt 复形之间的 p-adic Cartier 同构

A p-adic Cartier isomorphism between the A_inf-cohomology and de Rham-Witt complexes for semistable formal schemes

本文证明了半稳定形式概型上 A_inf-上同调与 de Rham-Witt 复形之间的 p-adic Cartier 同构，从而将 Bhatt-Morrow-Scholze 在光滑情形下的结果进行了推广。

Graded loopspaces in mixed characteristics and de Rham-Witt algebra

This PhD manuscript focuses on the study of a variation of the graded loop space construction for mixed graded derived schemes endowed with a Frobenius lift. We developed a theory of derived Frobenius lifts on a derived stack which are homotopy theoretic analogues of delta-structures for commutative rings. This graded loop space construction is the first step towards a definition of the de Rham-Witt complex for derived schemes. In this context, a loop is given by an action of the "crystalline circle", which is a formal analogue of the topological circle, endowed with its natural endomorphism g…

Gersten-Witt Complex of Hirzebruch Surfaces

Gersten-Witt cohomologies of Hirzebruch surfaces are computed.

光滑、无$\u200b\mathbb{Z}$-挠情形下相对德拉姆-维特复形的几何方法

A geometric approach to the relative de Rham-Witt complex in the smooth, $\mathbb{Z}$-torsion free case

本文研究有限生成平坦$\u200b\mathbb{Z}$-类代数上光滑概形的相对德拉姆-维特复形，证明其为对应奇异概形上通常德拉姆复形的无挠商，为该复形提供了几何视角。

Witt differentials in the h-topology

Recent important and powerful frameworks for the study of differential forms by Huber-Joerder and Huber-Kebekus-Kelly based on Voevodsky's h-topology have greatly simplified and unified many approaches. This article builds towards the goal of putting Illusie's de Rham-Witt complex in the same framework by exploring the h-sheafification of the rational de Rham-Witt differentials. Assuming resolution of singularities in positive characteristic one recovers a complete cohomological h-descent for all terms of the complex. We also provide unconditional h-descent for the global sections and draw the…

De Rham上同调与非Witt空间的分层解释

Stratified Interpretation for De Rham Cohomology and Non-Witt Spaces

本文建立层对应过滤空间与分层de Rham复形理论研究奇异空间，证明分层de Rham上同调与相交上同调的同构，并发展非Witt空间的分层结构理论，为奇异几何与拓扑不变量研究提供新基础。

de Rham-Witt上同调中的高维多米诺

Higher dimensional dominoes in de Rham-Witt cohomology

本文对de Rham-Witt上同调中的二维多米诺分类，将任意维多米诺标准化，赋予其两个幂幺群并证明同源划分一致，还计算超奇异阿贝尔簇多米诺、界定Brauer群指数，解答了相关问题。

通过范数构造的非交换相对德拉姆-维特复形

Noncommutative relative de Rham--Witt complex via the norm

该研究引入相对霍赫希尔德-维特同调，统一了德拉姆-维特复形的相对与非交换两种推广，证明其HKR定理并建立与相对德拉姆-维特复形的关联，还补全了Kaledin相关研究的缺失部分。

论p进解析空间的上同调，II：$C_{\ m st}$-猜想

On the cohomology of p-adic analytic spaces, II: The $C_{\rm st}$-conjecture

本文针对光滑dagger簇提出并证明Fontaine猜想的类比，统一了proper与Stein两种极端情形，核心方法是将相关上同调几何化为拟Banach-Colmez空间并利用其精细性质。

tame sheaves 和 tame de Rham--Witt cohomology 的构造

A construction of tame sheaves and tame de Rham--Witt cohomology

本文提出了一种构造 tame sheaves 的一般方法，应用于大 de Rham--Witt sheaves 和 reciprocity sheaves，并比较了 tame syntomic cohomology 与 tame de Rham--Witt 复合物上的 Nygaard 过滤。

$p$-adic vanishing cycles as Frobenius-fixed points

Given a smooth formal scheme over the ring of integers of a mixed-characteristic perfectoid field, we study its $p$-adic vanishing cycles via de Rham--Witt and $q$-de Rham complexes.

具有消没性的对数 de Rham-Witt 层的结构定理

Structure theorem for log de Rham-Witt sheaves with vanishing

本文证明了对数 de Rham-Witt 层沿有效 Cartier 除子消没的结构定理，回答了两位学者的问题，并为后续 Milnor K-理论研究奠定基础。

Some examples of DR-indecomposable special fibers of semi-stable reductions over Witt rings

We answer negatively an open problem of Illusie on the DR-decomposability of the log de Rham complex of the special fiber of a semi-stable reduction over the Witt ring. We also show that $E_1$ degeneration of the Hodge to log de Rham spectral sequence does not imply DR-decomposability of semi-stable varieties.

Equivariant splitting of the Hodge--de Rham exact sequence

Let $X$ be an algebraic curve with an action of a finite group $G$ over a field $k$. We show that if the Hodge-de Rham short exact sequence of $X$ splits $G$-equivariantly then the action of $G$ on $X$ is weakly ramified. In particular, this generalizes the result of Köck and Tait for hyperelliptic curves. We discuss also converse statements and tie this problem to lifting coverings of curves to the ring of Witt vectors of length $2$.

The algebraic de Rham cohomology of representation varieties

The SL(2,C)-representation varieties of punctured surfaces form natural families parameterized by holonomies at the punctures. In this paper, we first compute the loci where these varieties are singular for the cases of one-holed and two-holed tori and the four-holed sphere. We then compute the de Rham cohomologies of these varieties of the one-holed torus and the four-holed sphere when the varieties are smooth via the Grothendieck theorem. Furthermore, we produce the explicit Gauss-Manin connection on the natural family of the smooth SL(2,C)-representation variety of the one-holed torus.

$p$-adic Tate 扭曲上的 $p$-典型曲线与 de Rham--Witt 形式

$p$-typical curves on $p$-adic Tate twists and de Rham-Witt forms

本文研究 de Rham–Witt 形式与 p-adic Tate 扭曲上 p-典型曲线的关系；通过将 Bhatt–Morrow–Scholze 的动机滤子引入 Hesselholt 的拓扑循环同调结果，证明二者自然同构，回答了 Artin–Mazur 的长期问题。

通过de Rham-Witt复形的Kato分层及其应用

Kato's Ramification filtration via de Rham-Witt complex and applications

通过de Rham-Witt复形的分层，研究Kato分层的描述及其在对偶定理和Lefschetz定理中的应用。

De-Rham theorem and Shapiro lemma for Schwartz functions on Nash manifolds

In this paper we continue our work on Schwartz functions and generalized Schwartz functions on Nash (i.e. smooth semi-algebraic) manifolds. Our first goal is to prove analogs of de-Rham theorem for de-Rham complexes with coefficients in Schwartz functions and generalized Schwartz functions. Using that we compute cohomologies of the Lie algebra of an algebraic group G with coefficients in the space of generalized Schwartz sections of G-equivariant bundle over a G-transitive variety M. We do it under some assumptions on topological properties of G and M. This computation for the classical case i…

$q$-Witt向量与$q$-Hodge复形

$q$-Witt vectors and $q$-Hodge complexes

本文提出Witt向量和de Rham-Witt复形的$q$-变体，阐明其与Habiro环、$q$-Hodge上同调及$\text{THH}(-/\text{ku})$的关联，为后续相关研究提供技术基础。

具有对数极点的 TR 与 de Rham–Witt 复形

TR with logarithmic poles and the de Rham-Witt complex

本文研究 Hesselholt 关于对数 TR 与对数 de Rham–Witt 复形关系的猜想，在 p-adic 复数域上的 p-完全光滑形式概形情形中分析对数 TR^r 的动机滤过及 S^1-同伦不动点。

Mixed Weil cohomologies

We define, for a regular scheme $S$ and a given field of characteristic zero $\KK$, the notion of $\KK$-linear mixed Weil cohomology on smooth $S$-schemes by a simple set of properties, mainly: Nisnevich descent, homotopy invariance, stability (which means that the cohomology of $\GG_{m}$ behaves correctly), and Künneth formula. We prove that any mixed Weil cohomology defined on smooth $S$-schemes induces a symmetric monoidal realization of some suitable triangulated category of motives over $S$ to the derived category of the field $\KK$. This implies a finiteness theorem and a Poincaré dualit…

Cohomologie quantique orbifolde des espaces projectifs a poids

In 2001, S. Barannikov showed that the Frobenius manifold coming from the quantum cohomology of the complex projective space is isomorphic to the Frobenius manifold attached to some Laurent polynomial. The purpose of this thesis is to generalize this result. More precisely, we show, up to a conjecture on the value of some orbifold Gromov-Witten invariants, that the Frobenius structure obtained on the orbifold quantum cohomology of the weighted projective spaces is isomorphic to the one attached to some Laurent polynomial.

Vanishing theorems of Kodaira type for Witt canonical sheaves

Given a smooth projective variety over a perfect field of positive characteristic, we prove that the higher cohomologies vanish for the tensor product of the Witt canonical sheaf and the Teichmuller lift of an ample invertible sheaf. We also give a generalisation of this vanishing theorem to one of Kawamata-Viehweg type.

Stokes Matrices for the Quantum Cohomologies of Orbifold Projective Lines

We prove the Dubrovin's conjecture for the Stokes matrices for the quantum cohomology of orbifold projective lines. The conjecture states that the Stokes matrix of the first structure connection of the Frobenius manifold constructed from the Gromov-Witten theory coincides with the Euler matrix of a full exceptional collection of the bounded derived category of the coherent sheaves. Our proof is based on the homological mirror symmetry, primitive forms of affine cusp polynomials and the Picard-Lefschetz theory.

关于Tate Milnor-Witt Motives

On Tate Milnor-Witt Motives

该研究针对带孤立有理不动点的光滑射影$\u0001D706_m$-簇，证明其具有Tate Milnor-Witt motives，并在欧几里得域下给出分裂公式，简化Chow-Witt群的计算。

${\cal H}$-cohomologies versus algebraic cycles

Global intersection theories for smooth algebraic varieties via products in {\it appropriate}\, Poincaré duality theories are obtained. We assume given a (twisted) cohomology theory $H^*$ having a cup product structure and we let consider the ${\cal H}$-cohomology functor $X\leadsto H^{\#}_{Zar}(X,{\cal H}^*)$ where ${\cal H}^*$ is the Zariski sheaf associated to $H^*$. We show that the ${\cal H}$-cohomology rings generalize the classical ``intersection rings'' obtained via rational or algebraic equivalences. Several basic properties e.g.\, Gysin maps, projection formula and projective bundle …

驯顺上同调中的纯性

Purity in the tame cohomology

在特征p基概形上，假设奇点消解成立，证明了正则对(X,Z)的驯顺上同调中，对数de Rham-Witt层ν_m(n)的纯性，即自然同构Ri^!ν_{X,m}(n)≅ν_{Z,m}(n-r)[-r]的存在性。

Hodge-Witt decomposition of relative crystalline cohomology

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.

Witt微分算子

Witt Differential Operators

本文定义了正特征完美域上光滑概形的Witt向量上的微分算子环层，揭示了其与Berthelot环模及晶体范畴的深刻联系，发展了相应的拉回与前推形式体系，并推广了Bloch定理。