Bernhard Köck
We prove an equivariant Grothendieck-Ogg-Shafarevich formula. This formula may be viewed as an étale analogue of well-known formulas for Zariski sheaves generalizing the classical Chevalley-Weil formula. We give a new approach to those formulas (first proved by Ellingsrud/Lønsted, Nakajima, Kani and Ksir) which can also be applied in the étale case.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this article, we extend a pull-back inequality for total dimension divisors of étale sheavs due to Saito. Using this formula, we generalize Deligne and Laumon's lower semi-continuous property for Swan conductors of étale sheaves on relative curves to higher relative dimensions in a geometric situation.
Holonomic étale sheaves are constructible
基于贝林森工作，引入平展层全纯性概念，在完美基域上建立贝林森结果逆命题，证全纯层可构造，是柏原定理平展类似物。
A Microlocal Theory for Zariski-Constructible Sheaves on Rigid Analytic Varieties
本文在 Kashiwara–Schapira 框架下发展刚性解析簇上 Zariski 可构造层的微局部理论，引入单值层、Fourier 变换、特化、微局部化、micro-hom 与奇异支集等工具。
Vanishing Cycles for Zariski-Constructible Sheaves on Rigid Analytic Varieties
本文在非阿基米德域上建立了Zariski可构造层的近环与消失环理论，证明其保持可构造性、具有Milnor纤维解释、满足Beilinson粘合、perverse t-精确且与Verdier对偶交换。
Cohomologically smooth morphisms for étale $\mathbb F_p$-sheaves in characteristic $p$
本文研究了特征 $p$ 中平展 $\nathbb F_p$-层的六函子形式，证明上同调光滑态射仅由上同调平展态射构成，并刻画了后者在任意特征中实为平展态射（至普遍同胚）。
We prove a Zariski-Nagata purity theorem for the motivic ramification filtration of a reciprocity sheaf. An important tool in the proof is a generalization of the Kato-Saito reciprocity map from geometric global class field theory to all reciprocity sheaves. As a corollary we obtain cut-by-curves and cut-by-surfaces criteria for various ramification filtrations. In some cases this reproves known theorems, in some cases we obtain new results.
Morphing etale spaces
本文针对局部环几何单分支且亨泽尔的不可约概形，给出其小étale site上层范畴的简单描述，以此实现可表示层的刻画。
Let $X$ be an integral model at a prime $p$ of a Shimura variety of PEL type having good reduction, associated to a reductive group $G$. To $\mathbb{Z}_p$ reprsententations of the group $G$ can be associated two kinds of sheaves : crystals on the special fiber of $X$, and locally constant étale sheaves on the generic fiber. We establish a comparison between the cohomology of these two kinds of sheaves.
Remarks on logarithmic étale sheafification
本文研究对数概形上预层的对数平展层化问题，给出预层成为全对数平展层的判定准则，明确结构层与对数结构为对数平展层的情形，并证明满足对数平展下降的对数概形上对数皮卡群是全对数平展拓扑中的栈。
Quasi-étale covers of Du Val del Pezzo surfaces and Zariski dense exceptional sets in Manin's conjecture
本文构造了 Manin 猜想中具有 Zariski 稠密例外集的奇异 del Pezzo 曲面首批例子（次数为1、2和3），通过分类 Du Val del Pezzo 曲面的拟平展覆盖并研究群作用排除了高次情况，提供了系统构造方法。
In this article, we prove that the Swan conductor of an étale sheaf on a smooth variety defined by Abbes and Saito's logarithmic ramification theory can be computed by its classical Swan conductors after restricting it to curves. It extends the same result for rank 1 sheaves due to Barrientos. As an application, we give a logarithmic ramification version of generalizations of Deligne and Laumon's lower semi-continuity property for Swan conductors of étale sheaves on relative curves to higher relative dimensions in a geometric situation.
In this note we relate three topics for arithmetic schemes: a general duality for étale constructible torsion sheaves, an étale homology theory, and a Gersten-Bloch-Ogus-Kato complex. The results in this paper have been used in other papers of the authors ([JS], [Sa], [SaH] in the list of references).
We prove a generic smoothness result in rigid analytic geometry over a characteristic zero nonarchimedean field. The proof relies on a novel notion of generic points in rigid analytic geometry which are well-adapted to "spreading out" arguments, in analogy with the use of generic points in scheme theory. As an application, we develop a six functor formalism for Zariski-constructible étale sheaves on characteristic zero rigid spaces. Among other things, this implies that characteristic zero rigid spaces support a well-behaved theory of perverse sheaves.
In this paper, complement-equivalent arithmetic Zariski pairs will be exhibited answering in the negative a question by Eyral-Oka on these curves and their groups. A complement-equivalent arithmetic Zariski pair is a pair of complex projective plane curves having Galois-conjugate equations in some number field whose complements are homeomorphic, but whose embeddings in $\mathbb{P}^2$ are not. Most of the known invariants used to detect Zariski pairs depend on the étale fundamental group. In the case of Galois-conjugate curves, their étale fundamental groups coincide. Braid monodromy factorizat…
In this article the étale cohomology of constructible torsion sheaves on the étale site of the algebraic resp. adic Fargues-Fontaine curve is analyzed. In the $\ell\neq p$-torsion case, two conjectures of Fargues are verified: vanishing in degrees greater than two and the comparison between the étale cohomology of the adic and the algebraic curve. In the $p$-torsion case, under a certain assumption, the vanishing of the étale cohomology in degrees greater than two of those Zariski-constructible sheaves on the adic curve that come via pullback from the algebraic curve is proven.
On the stability of vanishing cycles of étale sheaves in positive characteristic
研究正特征下平展层消失圈的稳定性，表明光滑曲面上其对测试函数的依赖一般仅取决于有限阶喷流，猜想高维也成立，还研究了消失圈稳定性最强的层的类别及相关性质。
We define a theory of etale motives over a noetherian scheme. This provides a system of categories of complexes of motivic sheaves with integral coefficients which is closed under the six operations of Grothendieck. The rational part of these categories coincides with the triangulated categories of Beilinson motives (and is thus strongly related to algebraic $K$-theory). We extend the rigity theorem of Suslin and Voevodsky over a general base scheme. This can be reformulated by saying that torsion etale motives essentially coincide with the usual complexes of torsion etale sheaves (at least if…
In this article, we give a bound for the wild ramification of the monodromy action on the nearby cycles complex of a locally constant étale sheaf on the generic fiber of a smooth scheme over an equal characteristic trait in terms of Abbes and Saito's logarithmic ramification filtration. This provides a positive answer to the main conjecture in Isabel Leal's article "On the ramification of étale cohomology groups" for smooth morphisms in equal characteristic. We also study the ramification along vertical divisors of étale sheaves on relative curves and abelian schemes over a trait.
Recently, the singular support and the characteristic cycle of an étale sheaf on a smooth variety over a perfect field are constructed by Beilinson and Saito, respectively. In this article, we extend the singular support to a relative situation. As an application, we prove the generic constancy for singular supports and characteristic cycles of étale sheaves on a smooth fibration. Meanwhile, we show the failure of the lower semi-continuity of characteristic cycles in a higher relative dimension case, which is different from Deligne and Laumon's result in the relative curve case.
Let S be a Noetherian scheme and f:X -> S a proper morphism. By SGA 4 XIV, for any constructible sheaf F of Z/nZ-modules on X, the sheaves of Z/nZ-modules R^if_*F obtained by direct image (for the etale topology) are also constructible: there is a stratification of S on whose strata these sheaves are locally constant constructible. After previous work of N. Katz and G. Laumon, or L. Illusie, on the special case in which S is generically of characteristic zero or the sheaves F are constant (with invertible torsion on S), here we study the dependency of the stratification on F. We show that a na…
Semi-continuity for conductor divisors of étale sheaves
本文在几何情形下证明了高相对维数平展层的导子除子和对数导子除子的半连续性性质，推广了相对曲线平展层导子的半连续性结果，是André相关结果的高维ℓ-进类比。
Given a quasi-projective variety X with only Kawamata log terminal singularities, we study the obstructions to extending finite étale covers from the smooth locus $X_{\mathrm{reg}}$ of $X$ to $X$ itself. A simplified version of our main results states that there exists a Galois cover $Y \rightarrow X$, ramified only over the singularities of $X$, such that the étale fundamental groups of $Y$ and of $Y_{\mathrm{reg}}$ agree. In particular, every étale cover of $Y_{\mathrm{reg}}$ extends to an étale cover of $Y$. As first major application, we show that every flat holomorphic bundle defined on $…
We prove a purely local form of a result of Saito and Yatagawa. They proved that the characteristic cycle of a constructible étale sheaf is determined by wild ramification of the sheaf along the boundary of a compactification. But they had to consider ramification at all the points of the compactification. We give a pointwise result, that is, we prove that the characteristic cycle of a constructible étale sheaf around a point is determined by wild ramification at the point. The key ingredient is to prove that wild ramification of the stalk of the nearby cycle complex of a constructible étale s…
Computing the cohomology of constructible étale sheaves on curves
本文提出了一种计算不可约曲线上的构造层上同调复形的算法，该算法仅涉及有限群，并研究了其复杂性。
Constructible sheaves on toric varieties
本文针对环面簇上的可构造层范畴给出显式计算，既简化了复数域上Braden与Lunts的相关描述，又首次在正特征下对任意域分裂环面簇的驯分歧可构造平展层完成了平展外定理的显式计算。
Additivity of non-acyclicity classes for constructible étale sheaves
本文利用双变上同调对应为非反圈类建立范畴迹类公式，并以此证明其可加性。
Tame and curve-tame cohomology of $\mathbb A^1$-invariant étale sheaves
本文在附加假设下将非分歧曲线驯顺上同调推广至$\mathbb{A}^1$-不变平展层，定义其与苏斯林同调的配对，并证明正特征域上光滑几何连通簇的该上同调等于基域上同调。
The étale fundamental group and $F$-divided sheaves in characteristic $p>0$
本文证明特征 p 下光滑射影簇态射诱导的 étale 基本群满射或同构，可分别保证 F-divided 层拉回函子完全忠实或范畴等价，推广 Esnault–Mehta 定理。
A classic result by Raynaud and Gruson says that the notion of an (infinite dimensional) vector bundle is Zariski local. This result may be viewed as a particular instance (for n = 0) of the locality of more general notions of quasi-coherent sheaves related to (infinite dimensional) n-tilting modules and classes. Here, we prove the latter locality for all n and all schemes. We also prove that the notion of a tilting module descends along arbitrary faithfully flat ring morphisms in several particular cases (including the case when the base ring is noetherian).
For an $F$-finite scheme $X$ separated over a perfect field $k$ of characteristic $p>0$ which admits an embedding into a smooth $k$-scheme, we establish an equivalence between the bounded derived categories of Cartier crystals on $X$ and constructible $\mathbb{Z}/p\mathbb{Z}$-sheaves on the étale site $X_{\text{ét}}$. The key intermediate step is to extend the category of locally finitely generated unit $\mathcal{O}_{F,X}$-modules for smooth schemes introduced by Emerton and Kisin to embeddable schemes. On the one hand, this category is equivalent to Cartier crystals. On the other hand, by usi…
This paper continues our study of the sheaf associated to Kähler differentials in the cdh-topology and its cousins, in positive characteristic, without assuming resolution of singularities. The picture for the sheaves themselves is now fairly complete. We give a calculation $\mathcal{O}_{cdh}(X) \cong \mathcal{O}(X^{sn})$ in terms of the seminormalisation. We observe that the category of representable cdh-sheaves is equivalent to the category of seminormal varieties. We conclude by proposing some possible connections to Berkovich spaces, and $F$-singularities in the last section. The tools dev…
In this work we shall introduce a new model structure on the category of pro-simplicial sheaves, which is very convenient for the study of étale homotopy. Using this model structure we define a pro-space associated to a topos, as a result of applying a derived functor. We show that our construction lifts Artin and Mazur's étale homotopy type [AM] in the relevant special case. Our definition extends naturally to a relative notion, namely, a pro-object associated to a map of topoi. This relative notion lifts the relative étale homotopy type that was used in [HaSc] for the study of obstructions t…
A condensed proof of the pro-étale and étale exodromy theorems
本文通过简洁的方法证明了pro-étale和étale的exodromy定理，提出了一个新的关于Postnikov完备étalesheaves的exodromy定理，并给出了Barwick, Glasman和Haine的constructibleétaleexodromy对应关系的新证明，同时去除了对scheme的qcqs假设，扩展了sheaves的系数范围。
Suppose $X$ is a smooth, proper, geometrically connected curve over $\mathbb F_q$ with an $\mathbb F_q$-rational point $x_0$. For any $\mathbb F_q^{\times}$-character $σ$ of $π_1(X)$ trivial on $x_0$, we construct a functor $\mathbb L_n^σ$ from the derived category of coherent sheaves on the moduli space of deformations of $σ$ over the Witt ring $W_n(\mathbb F_q)$ to the derived category of constructible $W_n(\mathbb F_q)$-sheaves on the Jacobian of $X$. The functors $\mathbb L_n^σ$ categorify the Artin reciprocity map for geometric class field theory with $p$-torsion coefficients. We then giv…
We show that for quasi-compact quasi-separated schemes of finite dimension, the constructibility condition in real étale cohomology agrees with a notion of constructibility arising naturally from topology. As application we prove that the derived direct image functor preserves constructibility under some assumptions, and compute the Grothendieck group of the constructible rational stable motivic homotopy category for reasonable schemes. We prove the generic base change property for constructible real étale sheaves, and deduce the same property for rational motivic spectra and $b$-sheaves.
Grothendieck Galois theory and étale exodromy
针对原有 étale 外基本群定理证明冗长繁琐的问题，本文以 Grothendieck Galois 理论的风格，给出了固定分层下集合值可构造层的 étale 外基本群定理的简短证明。
In this paper we develop a theory of Grothendieck's six operations of lisse-étale constructible sheaves on Artin stacks which are locally of finite type over suitable regular basis of dimension at most 1.
We give a new geometric characterization of the motivic ramification filtration of reciprocity sheaves, by imitating a method used by Abbes and (Takeshi) Saito to study the ramification of torsors under finite étale groups. This new characterization is used to define characteristic forms for reciprocity sheaves. We obtain applications on pseudo-rational singularities and on questions regarding the representability of certain cohomology groups of reciprocity sheaves in the triangulated category of motives with modulus introduced by Kahn-Miyazaki-Saito-Yamazaki.
Weil-étale cohomology and the equivariant Tamagawa number conjecture for constructible sheaves in characteristic $p$
本文针对有限域上簇的可构造 étale 层，利用 Weil-étale 上同调证明了相关非交换 L-函数在负整数处的特殊值公式，从而给出等变玉川数猜想的几何类比并推广了若干已有结果。
We give a new definition of the derived category of constructible $\ell$-adic sheaves on a scheme, which is as simple as the geometric intuition behind them. Moreover, we define a refined fundamental group of schemes, which is large enough to see all lisse $\ell$-adic sheaves, even on non-normal schemes. To accomplish these tasks, we define and study the pro-étale topology, which is a Grothendieck topology on schemes that is closely related to the étale topology, and yet better suited for infinite constructions typically encountered in $\ell$-adic cohomology. An essential foundational result i…
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.
We introduce the characteristic class of an l-adic etale sheaf using a cohomological pairing due to Verdier (SGA5). As a consequence of the Lefschetz-Verdier trace formula, its trace computes the Euler-Poincare characteristic of the sheaf. We compare the characteristic class to two other invariants arising from ramification theory. One is the Swan class of Kato-Saito (math.AG/0402010) and the other is the 0-cycle class defined by Kato for rank 1 sheaves.
The Grothendieck-Ogg-Shafarevich formula expresses the Euler characteristic of an etale sheaf on a curve in terms of local data. The purpose of this paper is to prove a version of the G-O-S formula which applies to equicharacteristic sheaves (a bound, rather than an equality). This follows a proposal of R. Pink. The basis for the result is the characteristic-p "Riemann-Hilbert" correspondence, which relates equicharacteristic etale sheaves to O_{F, X}-modules. In the paper we prove a version of this correspondence for curves, considering both local and global settings. In the process we define…
A Uniform Construction of Cohomology Theories of Varieties via Fundamental Groupoids
本文提出了一种基于pro-代数基本群oid的代数上同调理论，通过替代传统上同调映射，为不同上同调理论提供统一基础。
Let X be an Enriques surface over the field of complex numbers. We prove that there exists a nontrivial quaternion algebra A on X. Then we study the moduli scheme of torsion free A-modules of rank one. Finally we prove that this moduli scheme is an étale double cover of a Lagrangian subscheme in the corresponding moduli scheme on the associated covering K3 surface.
In a previous paper, we constructed a category of (phi, Gamma)-modules associated to any adic space over Q_p with the property that the etale (phi, Gamma)-modules correspond to etale Q_p-local systems; these involve sheaves of period rings for Scholze's pro-etale topology. In this paper, we first extend Kiehl's theory of coherent sheaves on rigid analytic spaces to a theory of pseudocoherent sheaves on adic spaces, then construct a corresponding theory of pseudocoherent (phi, Gamma)-modules. We then relate these objects to a more explicit construction in case the space comes equipped with a su…
Let $F$ be a torsionfree semistable coherent sheaf on a polarized normal projective variety. We prove that $F$ has a unique maximal locally free subsheaf $V$ such that $F/V$ is torsionfree and $V$ admits a filtration of subbundles for which each successive quotient is stable whose slope is $μ(F)$. We also prove that $F$ has a unique maximal reflexive subsheaf $W$ such that $F/W$ is torsionfree and $W$ admits a filtration of subsheaves for which each successive quotient is a stable reflexive sheaf whose slope is $μ(F)$. We show that these canonical subsheaves behave well with respect to the pul…
This thesis deals with the algorithmic representation of constructible sheaves of abelian groups on the étale site of a variety over an algebraically closed field, as well as the explicit computation of their cohomology. We describe three representations of such sheaves on curves with at worst nodal singularities, as well as algorithms performing various operations (kernels and cokernels of morphisms, pullback and pushforward, internal Hom and tensor product) on these sheaves. We present an algorithm computing the cohomology complex of a locally constant constructible sheaf on a smooth or noda…
We prove a finiteness theorem and a comparison theorem in the theory of étale cohomology of rigid analytic varieties. By a result of Huber, for a quasi-compact separated morphism of rigid analytic varieties with target being of dimension $\le1$, the compactly supported higher direct image preserves quasi-constructibility. Though the analogous statement for morphisms with higher dimensional target fails in general, we prove that, in the algebraizable case, it holds after replacing the target with a modification. We deduce it from a known finiteness result in the theory of nearby cycles over gen…
Galois and Pro-étale Cohomology of Overconvergent de Rham Period Rings
受 p 进微分方程和 p 进几何表示论启发，引入 Fontaine 经典周期环的超收敛变体，计算其 Galois 上同调及相关周期层和周期结构层的上同调。
A 6-functor formalism for solid quasi-coherent sheaves on the Fargues-Fontaine curve
本文在小v-栈上建立了带$\mathbb{Z}_p$系数的6-函子形式体系，应用于刚性解析簇的pro-étale上同调的对偶与有限性，并给出局部Langlands对应$p$-进几何化纲领的首批例子。
There are two fundamental obstructions to representing noncommutative rings via sheaves. First, there is no subcanonical coverage on the opposite of the category of rings that includes all covering families in the big Zariski site. Second, there is no contravariant functor F from the category of rings to the category of ringed categories whose composite with the global sections functor is naturally isomorphic to the identity, such that F restricts to the Zariski spectrum functor Spec on the category of commutative rings (in a compatible way with the natural isomorphism). Both of these no-go re…
We construct the Weil-étale cohomology and Euler characteristics for a subclass of the class of $\mathbb{Z}$-constructible sheaves on the spectrum of the ring of integers of a totally imaginary number field. Then we show that the special value of an Artin L-function of toric type at zero is given by the Weil-étale Euler characteristic of an appropriate $\mathbb{Z}$-constructible sheaf up to signs. As applications of our result, we will derive some classical formulas of special values of L-functions of tori and their class numbers.
Unstable étale motives
本文在有界 étale 上同调维数的基概形上证明 anima 的 p-完备 étale A¹-不变层刚性，并用于构造不稳定 étale 实现函子。
We prove that the Brauer group of TMF is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Then, we compute the local Brauer group, i.e., the subgroup of the Brauer group of elements trivialized by some étale cover of the moduli stack, up to a finite 2-torsion group.
In this paper, given a smooth proper scheme X over a p-adic dvr and a p-power torsion etale local system L on it, we study a family of sheaves associated to the cohomology of local relative (Phi-Gamma)-modules of L and their cohomology. As applications we derive descriptions of the etale cohomology groups on the geometric generic fiber of X with values in L, as well as of their classical (Phi-Gamma)-modules, in terms of cohomology of the above mentioned sheaves.
Varieties with free tangent sheaves
提出T-平凡代数簇概念，研究特征p下其与阿贝尔簇的关联，证明可分支配的T-平凡簇仅在p≤3存在，还得到有限平展覆盖后的纤维化结果及提升特征零情形的覆盖结论。
We compute equivariant Euler characteristics of locally free sheaves on curves, thereby generalizing several results of Kani and Nakajima. For instance, we extend Kani's computation of the Galois module structure of the space of global meromorphic differentials which are logarithmic along the ramification locus from the tamely ramified to the weakly ramified case.
In 2010, Poonen gave the first example of failure of the local-global principle that cannot be explained by Skorobogatov's étale Brauer-Manin obstruction. Motivated by this example, we show that the Brauer-Manin obstruction detects non-existence of rational points on a sufficiently fine Zariski open cover of any variety over an imaginary quadratic or totally real field. We provide some evidence for why this is expected to happen more generally over any number field, some of which relates to the section conjecture in anabelian geometry. We then prove a result about the behavior of the étale Bra…
The Pro-Étale Homotopy Type
本文定义概形的pro-étale同伦型，证明其投射有限性等性质，计算特殊概形的该同伦型，还结合凝聚集等理论给出精细化定义以定义相关同伦群，推进了代数几何中同伦型理论的研究。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。