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Units of twisted group rings and their correlations to classical group rings 扭曲群环的单元及其与经典群环的关联

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-10-24 DOI: 10.1016/j.aim.2024.109983

Geoffrey Janssens , Eric Jespers , Ofir Schnabel

This paper is centred around the classical problem of extracting properties of a finite group G from the ring isomorphism class of its integral group ring

Z G

. This problem is considered via describing the unit group

U (Z G)

generically for a finite group. Since the ‘90s’ several well known generic constructions of units are known to generate a subgroup of finite index in

U (Z G)

Q G

does not have so-called exceptional simple epimorphic images, e.g.

M_{2} (Q)

. However it remained a major open problem to find a generic construction under the presence of the latter type of simple images. In this article we obtain such generic construction of units. Moreover, this new construction also exhibits new properties, such as providing generically free subgroups of large rank. As an application we answer positively for several classes of groups recent conjectures on the rank and the periodic elements of the abelianisation

U {(Z G)}^{a b}

. To obtain all this, we investigate the group ring RΓ of an extension Γ of some normal subgroup N by a group G, over a domain R. More precisely, we obtain a direct sum decomposition of the (twisted) group algebra of Γ over the fraction field F of R in terms of various twisted group rings of G over finite extensions of F. Furthermore, concrete information on the kernel and cokernel of the associated projections is obtained. Along the way we also launch the investigations of the unit group of twisted group rings and of

U (R Γ)

via twisted group rings.

本文围绕一个经典问题展开，即从有限群 G 的积分群环 ZG 的环同构类中提取有限群 G 的性质。这个问题是通过描述有限群的单位群 U(ZG) 来考虑的。自上世纪 90 年代以来，如果 QG 没有所谓的特殊简单外貌像（如 M2(Q)），已知的几种单位泛函构造可以在 U(ZG)中生成一个有限索引的子群。然而，在存在后一类简单映像的情况下，如何找到通用构造仍是一个重大的未决问题。在本文中，我们得到了这种单位的一般构造。此外，这种新构造还表现出新的特性，如提供大等级的泛自由子群。作为应用，我们正面回答了最近关于无秩化 U(ZG)ab 的秩和周期元素的几类群的猜想。为了实现这一切，我们研究了某个正则子群 N 由一个群 G 在一个域 R 上的扩展 Γ 的群环 RΓ。更确切地说，我们根据 G 在 F 的有限扩展上的各种扭曲群环，得到了 Γ 在 R 的分数域 F 上的（扭曲）群代数的直接和分解。同时，我们还通过扭曲群环展开了对扭曲群环的单位群和 U(RΓ) 的研究。

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引用次数: 0

Truncated pushforwards and refined unramified cohomology 截断前推和精制无ramified同调

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-10-23 DOI: 10.1016/j.aim.2024.109979

Theodosis Alexandrou , Stefan Schreieder

For a large class of cohomology theories, we prove that refined unramified cohomology is canonically isomorphic to the hypercohomology of a natural truncated complex of Zariski sheaves. This generalizes a classical result of Bloch and Ogus and solves a conjecture of Kok and Zhou.

对于一大类同调理论，我们证明了精炼的非ramified同调与扎里斯基剪切的自然截断复数的超同调是同构的。这概括了布洛赫和奥古斯的一个经典结果，并解决了郭和周的一个猜想。

引用次数: 0

Frobenius representation type for invariant rings of finite groups 有限群不变环的弗罗贝尼斯代表类型

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-10-23 DOI: 10.1016/j.aim.2024.109978

Mitsuyasu Hashimoto , Anurag K. Singh

Let V be a finite rank vector space over a perfect field of characteristic

p > 0

, and let G be a finite subgroup of

GL (V)

. If V is a permutation representation of G, or more generally a monomial representation, we prove that the ring of invariants

{(Sym V)}^{G}

has finite Frobenius representation type. We also construct an example with V a finite rank vector space over the algebraic closure of the function field

F_{3} (t)

, and G an elementary abelian subgroup of

GL (V)

, such that the invariant ring

{(Sym V)}^{G}

does not have finite Frobenius representation type.

设 V 是特征 p>0 的完全域上的有限秩向量空间，设 G 是 GL(V) 的有限子群。如果 V 是 G 的置换表示，或者更一般地说是单项式表示，我们将证明不变式环 (SymV)G 具有有限的弗罗贝尼斯表示类型。我们还构建了一个例子，V 是函数场 F3(t) 代数闭合上的有限秩向量空间，G 是 GL(V) 的基本无性子群，这样不变环 (SymV)G 就不具有有限弗罗贝尼斯表示类型。

引用次数: 0

Asymptotic behavior of complete conformal metric near singular boundary 奇异边界附近完全保角度量的渐近行为

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-10-21 DOI: 10.1016/j.aim.2024.109977

Weiming Shen, Yue Wang

The boundary behavior of the singular Yamabe problem has been extensively studied near sufficiently smooth boundaries, while less is known about the asymptotic behavior of solutions near singular boundaries. In this paper, we study the asymptotic behaviors of solutions to the singular Yamabe problem with negative constant scalar curvature near singular boundaries and derive the optimal estimates for the background metric which is not necessarily conformally flat. In particular, we prove that the solutions are well approximated by the solutions in tangent cones at singular points on the boundaries.

奇异山边问题的边界行为已在足够光滑的边界附近得到广泛研究，而对奇异边界附近解的渐近行为却知之甚少。本文研究了奇异边界附近具有负常标量曲率的奇异 Yamabe 问题解的渐近行为，并推导出不一定保角平坦的背景度量的最优估计。特别是，我们证明了边界上奇异点的解与切锥中的解近似得很好。

引用次数: 0

Milnor-Witt motivic cohomology and linear algebraic groups 米尔诺-维特动机同调与线性代数群

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-10-18 DOI: 10.1016/j.aim.2024.109973

Keyao Peng

This article presents two key computations in MW-motivic cohomology. Firstly, we compute the MW-motivic cohomology of the symplectic groups

{Sp}_{2 n}

for any

n \in N

using the Sp-orientation and the associated Borel classes.

Secondly, following the classical computations and using the analogue in

A^{1}

-homotopy of the Leray spectral sequence, we compute the η-inverted MW-motivic cohomology of general Stiefel varieties, obtaining in particular the computation of the η-inverted MW-motivic cohomology of the general linear groups

{GL}_{n}

and the special linear groups

{SL}_{n}

for any

n \in N

Finally, we determine the multiplicative structures of these total cohomology groups.

本文介绍了 MW 动同调的两个关键计算。首先，对于任意 n∈N 的交映群 Sp2n，我们利用 Sp 方向和相关的伯勒类计算其 MW 动同调。其次，我们按照经典计算方法，利用李雷谱序列在 A1-同调中的类比，计算了一般 Stiefel varieties 的 η-反转 MW-动同调，特别是计算了一般线性群 GLn 和特殊线性群 SLn 对于任意 n∈N 的 η-反转 MW-动同调。

引用次数: 0

Shuffle algebras, lattice paths and Macdonald functions 洗牌代数、格子路径和麦克唐纳函数

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-10-18 DOI: 10.1016/j.aim.2024.109974

Alexandr Garbali, Ajeeth Gunna

<div><div>We consider partition functions on the <math><mi>N</mi><mo>×</mo><mi>N</mi></math> square lattice with the local Boltzmann weights given by the R-matrix of the <math><msub><mrow><mi>U</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><msub><mrow><mover><mrow><mi>s</mi><mi>l</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>|</mo><mi>m</mi></mrow></msub><mo>)</mo></math> quantum algebra. We identify boundary states such that the square lattice can be viewed on a conic surface. The partition function <math><msub><mrow><mi>Z</mi></mrow><mrow><mi>N</mi></mrow></msub></math> on this lattice computes the weighted sum over all possible closed coloured lattice paths with <math><mi>n</mi><mo>+</mo><mi>m</mi></math> different colours: n “bosonic” colours and m “fermionic” colours. Each bosonic (fermionic) path of colour i contributes a factor of <math><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub></math> (<math><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi></mrow></msub></math>) to the weight of the configuration. We show the following:<ul><li>i)<div><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>N</mi></mrow></msub></math> is a symmetric function in the spectral parameters <math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>…</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>N</mi></mrow></msub></math> and generates basis elements of the commutative trigonometric Feigin–Odesskii shuffle algebra. The generating function of <math><msub><mrow><mi>Z</mi></mrow><mrow><mi>N</mi></mrow></msub></math> admits a shuffle-exponential formula analogous to the Macdonald Cauchy kernel.</div></li><li>ii)<div><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>N</mi></mrow></msub></math> is a symmetric function in two alphabets <math><mo>(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>…</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math> and <math><mo>(</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>…</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></math>. When <math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>…</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>N</mi></mrow></msub></math> are set to be equal to the box content of a skew Young diagram <math><mi>μ</mi><mo>/</mo><mi>ν</mi></math> with N boxes the partition function <math><msub><mrow><mi>Z</mi></mrow><mrow><mi>N</mi></mrow></msub></math> reproduces the skew Macdonald function <math><msub><mrow><mi>P</mi></mrow><mrow><mi>μ</mi><mo>/</mo><mi>ν</mi></mrow></msub><mrow><mo>[</mo><mi>w</mi><mo>−</mo><mi>z</mi><mo>]</mo></mrow></math>.</div></li></

我们考虑的是 N×N 方阵上的分割函数，其局部玻尔兹曼权重由 Ut(slˆn+1|m) 量子代数的 R 矩阵给出。我们确定了边界态，从而可以在圆锥面上观察方阵。该晶格上的分治函数 ZN 计算了所有可能的封闭彩色晶格路径的加权和，这些路径有 n+m 种不同颜色：n 种 "玻色 "和 m 种 "费米子 "色。每种颜色 i 的玻色（费米子）路径都会对配置的权重产生 zi (wi) 的影响。我们证明如下：i)ZN 是光谱参数 x1...xN 的对称函数，并生成交换三角费金-奥德斯基洗牌代数的基元。ZN 的生成函数有一个类似于 Macdonald Cauchy 核的洗牌-指数公式.ii)ZN 是两个字母表 (z1...zn) 和 (w1...wm) 中的对称函数。当 x1...xN 设为等于具有 N 个方框的倾斜杨图 μ/ν 的方框内容时，分割函数 ZN 重现了倾斜麦克唐纳函数 Pμ/ν[w-z]。

引用次数: 0

Stability of Llarull's theorem in all dimensions 拉鲁尔定理在所有维度上的稳定性

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-10-18 DOI: 10.1016/j.aim.2024.109980

Sven Hirsch , Yiyue Zhang

Llarull's theorem characterizes the round sphere

S^{n}

among all spin manifolds whose scalar curvature is bounded from below by

n (n - 1)

. In this paper we show that if the scalar curvature is bounded from below by

n (n - 1) - ε

, the underlying manifold is

C^{0}

-close to a finite number of spheres outside a small bad set. This completely solves Gromov's spherical stability problem and is the first instance of a scalar curvature stability result that both holds in all dimensions and is stated without any additional geometrical or topological assumptions.

拉鲁尔定理描述了所有标量曲率自下而上受 n(n-1) 约束的自旋流形中圆球 Sn 的特征。在本文中，我们证明了如果标量曲率自下而上受 n(n-1)-ε 约束，则底层流形在一个小的坏集之外与有限数量的球面是 C0-接近的。这完全解决了格罗莫夫的球面稳定性问题，是标量曲率稳定性结果的第一个实例，它既在所有维度上都成立，又无需任何额外的几何或拓扑假设。

引用次数: 0

Generalized cohomology theories for algebraic stacks 代数堆栈的广义同调理论

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-10-17 DOI: 10.1016/j.aim.2024.109975

Adeel A. Khan , Charanya Ravi

We extend the stable motivic homotopy category of Voevodsky to the class of scalloped algebraic stacks, and show that it admits the formalism of Grothendieck's six operations. Objects in this category represent generalized cohomology theories for stacks like algebraic K-theory, as well as new examples like genuine motivic cohomology and algebraic cobordism. These cohomology theories admit Gysin maps and satisfy homotopy invariance, localization, and Mayer–Vietoris. For example, we deduce that homotopy K-theory satisfies cdh descent on scalloped stacks. We also prove a fixed point localization formula for torus actions.

Finally, the construction is contrasted with a “lisse-extended” stable motivic homotopy category, defined for arbitrary stacks: we show for example that lisse-extended motivic cohomology of quotient stacks is computed by the equivariant higher Chow groups of Edidin–Graham, and we also get a good new theory of Borel-equivariant algebraic cobordism. Moreover, the lisse-extended motivic homotopy type is shown to recover all previous constructions of motives of stacks.

我们把沃沃茨基的稳定动机同调范畴扩展到扇形代数堆栈类，并证明它允许格罗内狄克的六次运算形式主义。这个范畴中的对象代表了堆栈的广义同调理论（如代数 K 理论），以及新的例子（如真正的动机同调和代数共线性）。这些同调理论承认Gysin映射，并满足同调不变性、局部性和Mayer-Vietoris。例如，我们推导出同调 K 理论在扇形堆栈上满足 cdh 下降。最后，我们将这一构造与针对任意堆栈定义的 "lisse-extended "稳定动机同构范畴进行了对比：例如，我们证明商堆栈的 lisse-extended 动机同构是由 Edidin-Graham 的等变高周群来计算的，我们还得到了一个很好的新的 Borel 等变代数共线性理论。此外，我们还证明了利塞扩展动机同调类型可以恢复以前所有的栈动机构造。

引用次数: 0

Duality for weak multiplier Hopf algebras with sufficiently many integrals 具有足够多积分的弱乘数霍普夫数组的对偶性

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-10-16 DOI: 10.1016/j.aim.2024.109971

Alfons Van Daele , Shuanhong Wang

We study duality of regular weak multiplier Hopf algebras with sufficiently many integrals. This generalizes the well-known duality of algebraic quantum groups. We need to modify the definition of an integral in this case. It is no longer true that an integral is automatically faithful and unique. Therefore we have to work with a faithful set of integrals. We apply the theory to three cases and give some examples. First we have the two weak multiplier Hopf algebras associated with an infinite groupoid (a small category). Related we answer a question posed by Nicolás Andruskiewitsch about double groupoids. Finally, we also discuss the weak multiplier Hopf algebras associated to a separability idempotent.

我们研究具有足够多积分的正则弱乘数霍普夫数组的对偶性。这概括了众所周知的代数量子群的对偶性。在这种情况下，我们需要修改积分的定义。积分自动忠实且唯一的说法不再成立。因此，我们必须使用一组忠实的积分。我们将这一理论应用于三种情况，并给出一些例子。首先是与无限群集（一个小范畴）相关的两个弱乘法霍普夫布拉斯。与此相关，我们回答了尼古拉斯-安德鲁斯基维奇（Nicolás Andruskiewitsch）提出的关于双群的问题。最后，我们还讨论了与可分离幂级数相关的弱乘数霍普夫数组。

引用次数: 0

The Koebe conjecture and the Weyl problem for convex surfaces in hyperbolic 3-space 双曲 3 空间凸面的科贝猜想和韦尔问题

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-10-15 DOI: 10.1016/j.aim.2024.109969

Feng Luo , Tianqi Wu

We prove that the Koebe circle domain conjecture is equivalent to the Weyl type problem that every complete hyperbolic surface of genus zero is isometric to the boundary of the hyperbolic convex hull of the complement of a circle domain in the hyperbolic 3-space. Applications of the result to discrete conformal geometry will be discussed. The main tool we use is Schramm's transboundary extremal lengths.

我们证明了 Koebe 圆域猜想等同于韦尔型问题，即每个零属的完整双曲面与双曲 3 空间中圆域补集的双曲凸壳边界等距。我们将讨论这一结果在离散共形几何中的应用。我们使用的主要工具是施拉姆的跨边界极值长度。

引用次数: 0

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