Akinari Hoshi, Ming-chang Kang, Aiichi Yamasaki
Let $p$ be an odd prime number. Peyre shows that there is a group $G$ of order $p^{12}$ such that $H_{nr}^3(\bm{C}(G), \bm{Q}/\bm{Z})$ is non-trivial. Using Peyre's method, we are able to prove that the same conclusion is true for some groups of order $p^9$.
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
We prove vanishing results for unramified stable cohomology of finite groups of Lie type.
We prove the universal triviality of the third unramified cohomology group of a very general complex cubic fourfold containing a plane. The proof uses results on the unramified cohomology of quadrics due to Kahn, Rost, and Sujatha.
This is a survey on unramified cohomology with a view towards its applications to rationality problems.
We construct and study universal spaces for birational invariants of algebraic varieties over algebraic closures of finite fields.
The purpose of this paper is to show that the third unramified cohomology with divisible coefficients of a smooth projective geometrically rational threefold over a finite field must vanish under $\Z_{\ell}$-exactness Hard Lefschetz condition.
Higher Chow groups with finite coefficients and refined unramified cohomology
本文证明域上拟射影等维概形的Bloch有限系数高阶圈类映射自然嵌入涉及Schreieder精细未分歧上同调的长正合序列，证明后者满足局部化序列，并猜想其为动机上同调理论。
On the functoriality of refined unramified cohomology
本文推广精细非分歧上同调的函子拉回与循环作用，证明射影丛、爆破公式，并将 Rost 幂零原理归结并验证于光滑真线性簇。
We develop an approach to proving the Rost nilpotence principle involving higher unramified cohomology. We use this to prove the principle for certain varieties of dimension $\leq 3$ over a perfect field.
Low degree unramified cohomology of generic diagonal hypersurfaces
本文证明在维数大于 i 的射影空间中，一般对角超曲面的第 1、2、3 次非分歧上同调群均为平凡群。
Combining work of Peyre, Colliot-Thélène and Voisin, we give the first example of a finite group $G$ such that the motivic class of its classifying stack $BG$ in Ekedahl's Grothendieck ring of stacks over $\mathbb{C}$ is non-trivial and $BG$ has trivial unramified Brauer group.
Truncated pushforwards and refined unramified cohomology
本文证明了精细化的非分歧上同调典范同构于Zariski层自然截断复形的超上同调，推广了Bloch-Ogus的经典结果并解决了Kok和Zhou的猜想。
We study the first "derived functors of unramified cohomology" in the sense of arXiv:1506.08385 [math.AG], applied to the sheaves $\mathbf{G}_m$ and $\mathcal{K}_2$. We find interesting connections with classical cycle-theoretic invariants of smooth projective varieties, involving notably a version of the Griffiths group, and the indecomposable $(2,1)$-cycles.
Auel-Bigazzi-Böhning-Graf von Bothmer proved that if a proper smooth variety $X$ over a field $k$ of characteristic $p>0$ has universally trivial Chow group of $0$-cycles, the cohomological Brauer group of $X$ is universally trivial as well. In this paper, we generalize their argument to arbitrary unramified mod $p$ étale motivic cohomology groups. We also see that the properness assumption on the variety $X$ can be dropped off by using the Suslin homology together with a certain tame subgroup of the unramified cohomology group.
Let BG be a classifying variety for an exceptional simple simply connected algebraic group G. We compute the degree 3 unramified Galois cohomology of BG with values in Q/Z(2) over an arbitrary field F. Combined with a paper by Merkurjev, this completes the computation of these cohomology groups for G semisimple simply connected over all fields. These computations provide another example of a simple simply connected group G such that BG is not stably rational.
On computation of unramified cohomology over non-closed fields
本文给出在非代数闭域上具有非平凡非分歧上同调的簇的例子，包括有界上同调维数域上的簇以及特征非2的任意域上有理曲面上的圆锥丛。
In this note, we consider unramified cohomology with $\mathbb{Z}/2$ coefficients for some (degree two) quotient varieties and describe a method that allows one to prove the non-vanishing of these groups under certain conditions. We apply this method to prove a non-vanishing statement in the case of Kummer varieties. Combining this with work of Colliot-Thélène and Voisin, we obtain a new type of three-dimensional counterexample to the integral Hodge conjecture.
We prove that the degree k unramified cohomology with torsion coefficients of a smooth complex projective variety X with small CH_0(X) has a filtration of length [k/2], whose first piece is the torsion part of the quotient of the degree k+1 integral singular cohomology by its coniveau 2 subgroup, and whose next graded piece is controlled by the Griffiths group Griff^{k/2+1}(X) when k is even and is related to the higher Chow group CH^{(k+3)/2}(X, 1) when k is odd. The first piece is a generalization of the Artin-Mumford invariant (k=2) and the Colliot-Thelene-Voisin invariant (k=3). We also gi…
We obtain several finiteness results for the unramified cohomology of function fields of algebraic varieties defined over fields of type (F'_m), a class that includes algebraically closed fields, finite fields, local fields, and some higher local fields of characteristic 0.
We study the Chevalley-Warning problem in the Grothendieck ring $K_0(Var/k)$. We show that the $\A^1$-homotopy theory yields well defined invariants on $K_0(Var/k)/\bold{L}$, in particular the Brauer group is such an invariant. We use this to give a concrete counter-example to the Chevalley-Warning conjecture over a $C_1$-field (Brown and Schnetz, 2011). This also gives a negative answer to the question in Bilgin (2011).
Unramified cohomology and Brauer--Manin pairing
本文证明局部域上光滑射影曲面非分歧上同调的Tate模在适当条件下消失，并应用于Brauer--Manin配对的核问题，通过建立Bloch公式实现。
We study Grothedieck groups of triangulated categories using weight structures, weight complexes, and the corresponding pure (co)homological functors. We prove some general statements on $K_0$ of weighted categories and apply it to Voevodsky motives endowed with so-called Chow weight structures. We obtain certain "motivic substitutes" for smooth compactifications of smooth varieties over arbitrary perfect fields; this enables us to make certain unramified cohomology and Euler characteristic calculations that are closely related to results of T. Ekedahl and B. Kahn.
Let $G$ be a finite group and $W$ be a faithful representation of $G$ over {\bf C}. The group $G$ acts on the field of rational functions $\mathbf C(W)$. The aim of this paper is to give a description of the unramified cohomology group of degree 3 of the field of invariant functions $\mathbf C(W)^G$ in terms of the cohomology of $G$ when $G$ is a group of odd order. This enables us to give an example of a group for which this field is not rational, although its unramified Brauer group is trivial.
This is a survey of recent examples of varieties that are not stably rational. We review the specialization method based on properties of the Chow group of zero-cycles used in these examples and explain the point of view of unramified cohomology for the construction of nontrivial stable invariants of the special fiber. In particular, we find an explicit formula for the Brauer group of fourfolds fibered in quadrics of dimension two over a rational surface.
Let $G$ be a reductive group defined over an algebraically closed field of characteristic $0$ such that the Dynkin diagram of $G$ is the disjoint union of diagrams of types $G_{2}, F_{4}, E_{6}, E_{7}, E_{8}$. We show that the degree $3$ unramified cohomology of the classifying space of $G$ is trivial. In particular, combined with articles by Merkurjev \cite{Mer17} and the author \cite{Baek}, this completes the computations of degree $3$ unramified cohomology and reductive invariants for all split semisimple groups of a homogeneous Dynkin type.
Unramified Cohomology of Quadrics in Characteristic Two
本文研究特征2域上光滑射影二次曲面的2、3次2-准素挠非分歧上同调，完全确定基域上同调到其非分歧上同调自然映射的核与余核，推广了特征非2情形的经典结果。
We formulate and analyze several finiteness conjectures for linear algebraic groups over higher-dimensional fields. In fact, we prove all of these conjectures for algebraic tori as well as in some other situations. This work relies in an essential way on several finiteness results for unramified cohomology.
Let $X$ be a smooth proper variety over a field $k$ and suppose that the degree map $\mathrm{CH}_0(X \otimes_k K) \to \mathbb{Z}$ is isomorphic for any field extension $K/k$. We show that $G(\mathrm{Spec} k) \to G(X)$ is an isomorphism for any $\mathbb{P}^1$-invariant Nisnevich sheaf with transfers $G$. This generalize a result of Binda-Rülling-Saito that proves the same conclusion for reciprocity sheaves. We also give a direct proof of the fact that the unramified logarithmic Hodge-Witt cohomology is a $\mathbb{P}^1$-invariant Nisnevich sheaf with transfers.
Vanishing of local cohomology in unramified mixed characteristic
本文在非分歧混合特征环境下扩展了 Faltings 关于理想上同调维数的上界结果，并证明该上界是紧致的。
Torsion birational motives of surfaces and unramified cohomology
该研究探讨了曲面的torsion birational motives与unramified cohomology的关系，并扩展了Kahn的结果，同时给出了违反积分Hodge猜想的例子。
In [Inventiones mathematicae, 184 (2011)], Vollaard and Wedhorn defined a stratification on the special fiber of the unitary unramified PEL Rapoport-Zink space with signature $(1,n-1)$. They constructed an isomorphism between the closure of a stratum, called a closed Bruhat-Tits stratum, and a Deligne-Lusztig variety which is not of classical type. In this paper, we describe the $\ell$-adic cohomology groups over $\overline{\mathbb Q_{\ell}}$ of these Deligne-Lusztig varieties, where $\ell \not = p$. The computations involve the spectral sequence associated to the Ekedahl-Oort stratification o…
We study the $l$-adic cohomology of unramified Rapoport-Zink spaces of EL-type. These spaces were used in Harris and Taylor's proof of the local Langlands correspondence for $\mathrm{GL_n}$ and to show local-global compatibilities of the Langlands correspondence. In this paper we consider certain morphisms, $\mathrm{Mant}_{b, μ}$, of Grothendieck groups of representations constructed from the cohomology of the above spaces, as studied by Harris and Taylor, Mantovan, Fargues, Shin, and others. Due to earlier work of Fargues and Shin we have a description of $\mathrm{Mant}_{b, μ}(ρ)$ for $ρ$ a s…
We introduce the notion of refined unramified cohomology of algebraic schemes and prove comparison theorems that identify some of these groups with cycle groups. This recovers for cycles of low codimensions on smooth projective varieties previous results of Bloch--Ogus, Colliot-Thélène--Voisin, Kahn, Voisin, and Ma. We combine our approach with the Bloch--Kato conjecture, proven by Voevodsky, to show that on a smooth complex projective variety, any homologically trivial torsion cycle with trivial Abel--Jacobi invariant has coniveau 1. This establishes a torsion version of a conjecture of Janns…
Let $k$ be a field and $G$ be a finite group acting on the rational function field $k(x_g : g\in G)$ by $k$-automorphisms defined as $h(x_g)=x_{hg}$ for any $g,h\in G$. We denote the fixed field $k(x_g : g\in G)^G$ by $k(G)$. Noether's problem asks whether $k(G)$ is rational (= purely transcendental) over $k$. It is well-known that if $C(G)$ is stably rational over $C$, then all the unramified cohomology groups $H_[nr}^i(C(G),Q/Z)=0$ for $i \ge 2$. Hoshi, Kang and Kunyavskii [HKK] showed that, for a $p$-group of order $p^5$ ($p$: an odd prime number), $H_[nr}^2(C(G),Q/Z)\neq 0$ if and only if …
Vanishing of degree $3$ unramified cohomology over finite fields
本文证明有限域上费马三次曲线三次非分岐上同调消没及循环映射满射，并推广至满足 Tate 猜想的簇，应用于零循环局部-全局原理。
Finiteness of associated primes for local cohomology modules of excellent locally unramified regular rings of finite Krull dimension
本文证明了Huneke和Lyubeznik关于局部上同调模相伴素理想有限性的猜想，在有限Krull维数的优秀局部无分歧正则环等条件下取得了实质性新进展。
A method of Gabber (2002) produces unramified cohomology classes in the products of certain varieties with an elliptic curve. The connection between third unramified cohomology and integral Hodge conjecture for codimension 2 cycles (2012, which builds upon results from algebraic K-Theory, then gives many examples of such a product for which this conjecture fails. The special case of the product with an Enriques surface was established by Benoist and Ottem (2018).
Separating components of real algebraic varieties by cohomological invariants
该研究针对实数域上的光滑代数簇，利用等变上同调、非分歧上同调及非分歧二次型符号分离其实点连通分支，将结果应用于多种代数簇并得到$\boldsymbol{I}$-上同调相关新界及最优性结论。
Using the notion of isoclinism introduced by P. Hall for finite p-groups, we show that many important classes of finite p-groups have stable cohomology detected by abelian subgroups, see Theorem 4.4. Moreover, we show that the stable cohomology of the n-fold wreath product of cyclic groups ZZ/p is detected by elementary abelian p-subgroups and we describe the resulting cohomology algebra explicitly. Some applications to the computation of unramified and stable cohomology of finite groups of Lie type are given.
A coproduct obstruction for derived unramified cohomology
本文证明在指数特征可逆后，第一个导出非分支函子与任意小直和交换，并描述第二个导出非分支函子的范畴障碍，应用于曲面的扭阶界和对应检测。
This is a textbook on arithmetic geometry with special regard to unramified Brauer groups of algebraic varieties. The topics include Galois cohomology, Brauer groups, obstructions to stable rationality, arithmetic and geometry of quadrics, Weil restriction of scalars, algebraic tori, an example of a stably rational non-rational variety, Brauer-Manin obstruction. All material is split into locally trivial problems with detailed hints.
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。
We develop birational versions of Voevodsky's triangulated categories of motives over a field, and relate them with the pure birational motives studied in arXiv:0902.4902 [math.AG]. We also get an interpretation of unramified cohomology in this framework, leading to "higher derived functors of unramified cohomology".
Azumaya algebras over unramified extensions of function fields
本文利用Milnor-Quillen $K$-群解释函数域的étale上同调挠部分，证明了在范数映射的温和条件下，光滑簇上存在可由Azumaya代数表示的上同调Brauer类，从而部分回答了Grothendieck的问题。
On the failure of the integral Hodge/Tate conjecture for products with projective hypersurfaces
本文通过细化无 ramifications cohomology方法，证明了Enriques面与光滑奇维投影超曲面的乘积上积分Hodge/塔特猜想的失效，并推广了Shen的结果到一般代数闭域。
Using a reduction of the Galois cohomology of a linear algebraic group $G$ to that of a certain finite subquotient, we give different formulas allowing the calculation of the unramified algebraic Brauer group of a homogeneous space $V=G\backslash G'$ with $G'$ semisimple and simply connected, both over a finite field and over an arbitrary field of characteristic $0$.
We relate the torsion part of the Abel-Jacobi kernel in the Griffiths group of 1-cycles to a birational invariant analogous to the degree 4 unramified cohomology and an invariant associated to the generalized Hodge conjecture in degree 2dim(X)-3. We also describe in terms of H-cohomology the Griffiths group of 1-cycles and the group of torsion cycles algebraically equivalent to zero of arbitrary dimension.
Let $X \subset \mathbb{P}^{n}$ be an unramified real curve with $X(\mathbb{R}) \neq \emptyset$. If $n \geq 3$ is odd, Huisman conjectures that $X$ is an $M$-curve and that every branch of $X(\mathbb{R})$ is a pseudo-line. If $n \geq 4$ is even, he conjectures that $X$ is a rational normal curve or a twisted form of a such. We disprove the first conjecture by giving a family of counterexamples. We remark that the second conjecture follows for generic curves of odd degree from the formula enumerating the number of complex inflection points.
Let K be a finite extension of Q_p and X a smooth projective variety over K. We define the notion of totally degenerate reduction of such an X and the associated Chow complexes of the special fibre of a suitable regular proper model of X over the ring of integers of K. If X has such reduction, we then show that for all l, the Q_l-adic etale cohomology of X has a filtration whose graded quotients are isomorphic, as Galois modules, to the tensor product of a finite dimensional Q-vector space (with a finite unramified action of Galois) with twists of Q_l by the cyclotomic character.
On the topology of the moduli of tropical unramified p-covers
本文研究亏格g≥2热带曲线非分歧Z/p-覆叠模空间的拓扑，利用Chan–Galatius–Payne技巧证明其单连通性，并在亏格2时确定所有素数p下的同伦型，为素数阶循环étale覆叠的顶权上同调提供应用前景。
A Néron-Ogg-Shafarevich criterion for K3 surfaces
本文针对K3曲面修正了内龙-奥格-沙法列维奇准则，在潜在半稳定约化假设下给出K3曲面具有好约化的等价条件，并证明了p进平展上同调的对应结果。
Let $F$ be a totally real field in which $p$ is unramified. We prove that, if a cuspidal overconvergent Hilbert cuspidal form has small slopes under $U_p$-operators, then it is classical. Our method follows the original cohomological approach of Coleman. The key ingredient of the proof is giving an explicit description of the Goren-Oort stratification of the special fiber of the Hilbert modular variety. A byproduct of the proof is to show that, at least when $p$ is inert, of the rigid cohomology of the ordinary locus has the same image as the classical forms in the Grothendieck group of Hecke …
We define formal vector bundles with marked sections on Hilbert modular schemes and we show how to use them to construct modular sheaves with an integrable meromorphic connection and a filtration which, in degree 0, gives to us a $p$-adic interpolation of the usual Hodge filtration. We relate it with the sheaf of overconvergent Hilbert modular forms.
Arithmetic jet spaces and the unramified Zilber--Pink conjecture for subvarieties of abelian varieties
该研究运用δ几何方法，针对定义在数域上的阿贝尔簇子簇，结合固定素数处有界分歧的点，界定其Zilber–Pink轨迹中点的次数，推进了非分歧Zilber–Pink猜想的相关研究。
We establish a "second vanishing theorem" for local cohomology modules over regular rings of unramified mixed characteristic, which relates the connectedness of the spectrum of a ring with the vanishing of local cohomology. Applying this, and new results on the mixed characteristic Lyubeznik numbers, we further study connectedness properties of the spectra of a certain class of rings.
Let $(\mathsf{G},\mathsf{X})$ be a Shimura datum of Hodge type. Let $p$ be an odd prime such that $\mathsf{G}_{\mathbb{Q}_p}$ splits after a tamely ramified extension and $p\nmid |π_1(\mathsf{G}^{\rm der})|$. Under some mild additional assumptions that are satisfied if the associated Shimura variety is proper and $\mathsf{G}_{\mathbb{Q}_p}$ is either unramified or residually split, we prove the generalisation of Mantovan's formula for the $l$-adic cohomology of the associated Shimura variety. On the way we derive some new results about the geometry of the Newton stratification of the reduction…
We determine the group of reductive cohomological degree $3$ invariants of all split semisimple groups of types $B$, $C$, and $D$. We also present a complete description of the cohomological invariants. As an application, we show that the group of degree $3$ unramified cohomology of the classifying space $BG$ is trivial for all split semisimple groups $G$ of types $B$, $C$, and $D$.
Let G be a finite group of exponent m and let k be a field of characteristic prime to m, containing the m-th roots of unity. For any Rost cycle module M over k, we construct exact sequences detecting the unramified elements in Serre's group of invariants of G with values in M in terms of "residue" morphisms associated to pairs (D,g), where D runs through the subgroups of G and g runs through the homomorphisms μ_m \to G whose image centralises D. This allows us to recover results of Bogomolov and Peyre on the unramified cohomology of fields of invariants of G, and to generalise them.
We use Burgos theory of arithmetic Chow groups to exhibit a realization of the degree zero part of the polylogarithm on abelian schemes in Deligne-Beilinson cohomology.
The eigenvalues of Frobenius acting on $\ell$-adic cohomology of a complete intersection over a finite field have the divisibility predicted by the theorem of Ax and Katz. We have corrected some unforgivable typos.
We show that the cohomology of the perfect cone (also called first Voronoi) toroidal compactification of the moduli space of complex principally polarized abelian varieties stabilizes, in close to the top degree. Moreover, we show that this stable cohomology is purely algebraic, and we compute it in degree up to 13. Our explicit computations and stabilization results apply in greater generality to various toroidal compactifications and partial compactifications, and in particular we show that the cohomology of the matroidal partial compactification stabilizes (in low degree). For degree up to …
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。