Brauer groups and étale homotopy type

Mohammed Moutand

备注：: 27 pages, comments are welcome

英文摘要：

Extending a result of Schröer on a Grothendieck question in the context of complex analytic spaces, we prove that the surjectivity of the Brauer map $δ: Br(X) \rightarrow H_{\rm ét}^2(X,\mathbb{G}_{m, X})_{\rm tor}$ for algebraic schemes depends on their étale homotopy type. We use properties of algebraic $K(π, 1)$ spaces to apply this to some classes of proper and smooth algebraic schemes. In particular we recover a result of Hoobler and Berkovich for abelian varieties. Further, we give an additional condition for the surjectivity of $δ$ which involves pro-universal covers. All proposed conditions turn out to be equivalent for smooth quasi-projective varieties.

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。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

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

相关论文

Homotopy Rational Points of Brauer-Severi Varieties

We study homotopy rational points of Brauer-Severi varieties over fields of characteristic zero. We are particularly interested if a Brauer-Severi variety admitting a homotopy rational point splits. The analogue statement turns out to be true for open subvarieties of twisted hyperplane arrangements. In the complete case, the obstruction turns out to be the pullback of the Chern class of a generator of the Picard group along the homotopy rational point. Moreover, we will show that every Brauer-Severi variety over a $p$-adic field with $p$-primary period admits a homotopy rational point.

2015-03-27math.AG

Brauer and Etale Homotopy Obstructions to Rational Points on Open Covers

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…

2020-08-16math.AGmath.NT

Duality for cohomology of split tori on curves over local fields

We prove duality theorems for the {é}tale cohomology of logarithmic Hodge-Witt sheaves and split tori on smooth curves over a local field of positive characteristic. As an application, we obtain a description of the Brauer group of the function fields of curves over local fields in terms of the characters of the idele groups. We also show that the classical Brauer-Manin pairing between the Brauer and Picard groups of smooth projective curves over local fields has analogues for arbitrary smooth curves, smooth projective curves with modulus and singular projective curves over such fields.

2023-02-13math.AG

Topological methods for complex-analytic Brauer groups

Using methods from algebraic topology and group cohomology, I pursue Grothendieck's question on equality of geometric and cohomological Brauer groups in the context of complex-analytic spaces. The main result is that equality holds under suitable assumptions on the fundamental group and the Pontrjagin dual of the second homotopy group. I apply this to Lie groups, Hopf manifolds, and complex-analytic surfaces.

2005-02-08math.AG

The Étale Homotopy Type and Obstructions to the Local-Global Principle

In 1969 Artin and Mazur defined the étale homotopy type of an algebraic variety \cite{AMa69}. In this paper we define various obstructions to the local-global principle on a variety $X$ over a global field using the étale homotopy type of $X$ and the concept of homotopy fixed points. We investigate relations between those "homotopy obstructions" and connect them to various known obstructions such as the Brauer -Manin obstruction, the étale-Brauer obstruction and finite descent obstructions. This gives a reinterpretation of known arithmetic obstructions in terms of homotopy theory.

2011-11-30math.AGmath.AT

Homotopy Obstructions to Rational Points

In this paper we propose to use a relative variant of the notion of the étale homotopy type of an algebraic variety in order to study the existence of rational points on it. In particular, we use an appropriate notion of homotopy fixed points in order to construct obstructions to the local-global principle. The main results in this paper are the connections between these obstructions and the classical obstructions, such as the Brauer-Manin, the étale-Brauer and certain descent obstructions. These connections allow one to understand the various classical obstructions in a unified framework.

2011-10-02math.AGmath.AT

更多相关论文收起更多论文

关于具有平凡高维 étale 同调群的方案

On schemes with trivial higher étale homotopy groups

Moulay Ismail University（穆莱伊斯梅尔大学）

本文通过 étale 同调类型和 universal 覆盖空间特征化了具有平凡高维 étale 同调群的方案，并给出了 Brauer 映射满射的条件。

2026-02-26math.AG

$\mathbb{A}^1$-连通叠曲线与椭圆曲线模空间的 Brauer 群

$\mathbb{A}^1$-connected stacky curves and the Brauer group of moduli of elliptic curves

本文提出商叠动机连通分支 Nisnevich 层的计算公式，据此分类 $\mathbb{A}^1$-连通叠曲线并证明椭圆曲线模叠的 $\mathbb{A}^1$-连通性，同时建立光滑代数叠同伦纯性并恢复 Picard 群计算。

2024-12-06math.AG

Brauer groups and étale cohomology in derived algebraic geometry

In this paper, we study Azumaya algebras and Brauer groups in derived algebraic geometry. We establish various fundamental facts about Brauer groups in this setting, and we provide a computational tool, which we use to compute the Brauer group in several examples. In particular, we show that the Brauer group of the sphere spectrum vanishes, and we use this to prove two uniqueness theorems for the stable homotopy category. Our key technical results include the local geometricity, in the sense of Artin n-stacks, of the moduli space of perfect modules over a smooth and proper algebra, the étale l…

2012-10-01math.AGmath.ATmath.RA