Finite torsors on projective schemes defined over a discrete valuation ring

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2019-04-24 DOI:10.14231/ag-2023-001

P. H. Hai, J. Santos

引用次数: 3

Abstract

Given a Henselian and Japanese discrete valuation ring $A$ and a flat and projective $A$-scheme $X$, we follow the approach of Biswas-dos Santos to introduce a full subcategory of coherent modules on $X$ which is then shown to be Tannakian. We then prove that, under normality of the generic fibre, the associated affine and flat group is pro-finite in a strong sense (so that its ring of functions is a Mittag-Leffler $A$-module) and that it classifies finite torsors $Q\to X$. This establishes an analogy to Nori's theory of the essentially finite fundamental group. In addition, we compare our theory with the ones recently developed by Mehta-Subramanian and Antei-Emsalem-Gasbarri. Using the comparison with the former, we show that any quasi-finite torsor $Q\to X$ has a reduction of structure group to a finite one.

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离散赋值环上定义的投影方案上的有限扭算子

给定一个Henselian和Japanese离散估值环$ a $和一个平面和投影的$ a $-方案$X$，我们遵循Biswas-dos Santos的方法，引入$X$上的相干模的完整子范畴，然后证明它是Tannakian的。然后证明了在一般纤维的正规性下，相关联的仿射平群在强意义上是亲有限的(因此它的函数环是一个Mittag-Leffler模)，并证明了它对有限环子$Q\到X$进行分类。这建立了与Nori关于本质上有限基本群的理论的类比。此外，我们将我们的理论与Mehta-Subramanian和Antei-Emsalem-Gasbarri最近发展的理论进行了比较。通过与前者的比较，我们证明了任意拟有限扭量$Q\to X$都有一个结构群约简为有限结构群。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.