scholarly journals MODELS OF TORSORS AND THE FUNDAMENTAL GROUP SCHEME

2016 ◽  
Vol 230 ◽  
pp. 18-34 ◽  
Author(s):  
MARCO ANTEI ◽  
MICHEL EMSALEM
Keyword(s):  
Finite Number ◽  
Fundamental Group ◽  
Galois Group ◽  
Closed Subset ◽  
Valuation Ring ◽  
Group Scheme ◽  
Discrete Valuation Ring ◽  
Generic Fiber ◽  
Natural Morphism ◽  
Tannakian Category

Given a relative faithfully flat pointed scheme over the spectrum of a discrete valuation ring $X\rightarrow S$, this paper is motivated by the study of the natural morphism from the fundamental group scheme of the generic fiber $X_{\unicode[STIX]{x1D702}}$ to the generic fiber of the fundamental group scheme of $X$. Given a torsor $T\rightarrow X_{\unicode[STIX]{x1D702}}$ under an affine group scheme $G$ over the generic fiber of $X$, we address the question of finding a model of this torsor over $X$, focusing in particular on the case where $G$ is finite. We provide several answers to this question, showing for instance that, when $X$ is integral and regular of relative dimension 1, such a model exists on some model $X^{\prime }$ of $X_{\unicode[STIX]{x1D702}}$ obtained by performing a finite number of Néron blowups along a closed subset of the special fiber of $X$. Furthermore, we show that when $G$ is étale, then we can find a model of $T\rightarrow X_{\unicode[STIX]{x1D702}}$ under the action of some smooth group scheme. In the first part of the paper, we show that the relative fundamental group scheme of $X$ has an interpretation as the Tannaka Galois group of a Tannakian category constructed starting from the universal torsor.

scholarly journals Higgs bundles and fundamental group schemes

2019 ◽  
Vol 19 (3) ◽  
pp. 381-388
Author(s):  
Indranil Biswas ◽  
Ugo Bruzzo ◽  
Sudarshan Gurjar
Keyword(s):  
Fundamental Group ◽  
Vector Bundles ◽  
Projective Variety ◽  
Group Scheme ◽  
Smooth Projective Variety ◽  
Higgs Bundles ◽  
Chern Classes ◽  
Group Schemes ◽  
Tannakian Category

Abstract Relying on a notion of “numerical effectiveness” for Higgs bundles, we show that the category of “numerically flat” Higgs vector bundles on a smooth projective variety X is a Tannakian category. We introduce the associated group scheme, that we call the “Higgs fundamental group scheme of X,” and show that its properties are related to a conjecture about the vanishing of the Chern classes of numerically flat Higgs vector bundles.

scholarly journals -neighborhoods and comparison theorems

2015 ◽  
Vol 151 (10) ◽  
pp. 1945-1964 ◽  
Author(s):  
Piotr Achinger
Keyword(s):  
Valuation Ring ◽  
Open Neighborhood ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Universal Cover ◽  
Comparison Theorems ◽  
Main Ingredient ◽  
Generic Fiber ◽  
Mixed Characteristic ◽  
Original Approach

A technical ingredient in Faltings’ original approach to$p$-adic comparison theorems involves the construction of$K({\it\pi},1)$-neighborhoods for a smooth scheme$X$over a mixed characteristic discrete valuation ring with a perfect residue field: every point$x\in X$has an open neighborhood$U$whose generic fiber is a$K({\it\pi},1)$scheme (a notion analogous to having a contractible universal cover). We show how to extend this result to the logarithmically smooth case, which might help to simplify some proofs in$p$-adic Hodge theory. The main ingredient of the proof is a variant of a trick of Nagata used in his proof of the Noether normalization lemma.

scholarly journals The Fundamental Group of a Noncommutative Space

2021 ◽  
Author(s):  
Walter D. van Suijlekom ◽  
Jeroen Winkel
Keyword(s):  
Fundamental Group ◽  
Classical Case ◽  
Group Scheme ◽  
Noncommutative Space ◽  
General Notion ◽  
Noncommutative Torus ◽  
Graded Algebras ◽  
Differential Graded Algebra ◽  
Differential Graded Algebras ◽  
Tannakian Category

AbstractWe introduce and analyse a general notion of fundamental group for noncommutative spaces, described by differential graded algebras. For this we consider connections on finitely generated projective bimodules over differential graded algebras and show that the category of flat connections on such modules forms a Tannakian category. As such this category can be realised as the category of representations of an affine group scheme G, which in the classical case is (the pro-algebraic completion of) the usual fundamental group. This motivates us to define G to be the fundamental group of the noncommutative space under consideration. The needed assumptions on the differential graded algebra are rather mild and completely natural in the context of noncommutative differential geometry. We establish the appropriate functorial properties, homotopy and Morita invariance of this fundamental group. As an example we find that the fundamental group of the noncommutative torus can be described as the algebraic hull of the topological group $(\mathbb Z+\theta \mathbb Z)^{2}$ ( ℤ + 𝜃 ℤ ) 2 .

scholarly journals On the Fundamental Group Scheme of Rationally Chain-Connected Varieties

2015 ◽  
Vol 2016 (1) ◽  
pp. 311-324
Author(s):  
Marco Antei ◽  
Indranil Biswas
Keyword(s):  
Fundamental Group ◽  
Group Scheme

scholarly journals Vector bundles trivialized by proper morphisms and the fundamental group scheme

2010 ◽  
Vol 10 (2) ◽  
pp. 225-234 ◽  
Author(s):  
Indranil Biswas ◽  
João Pedro P. Dos Santos
Keyword(s):  
Finite Group ◽  
Fundamental Group ◽  
Characteristic Zero ◽  
Vector Bundles ◽  
Projective Variety ◽  
Group Scheme ◽  
Smooth Projective Variety ◽  
Closed Field ◽  
Surjective Morphism ◽  
Finite Vector

AbstractLet X be a smooth projective variety defined over an algebraically closed field k. Nori constructed a category of vector bundles on X, called essentially finite vector bundles, which is reminiscent of the category of representations of the fundamental group (in characteristic zero). In fact, this category is equivalent to the category of representations of a pro-finite group scheme which controls all finite torsors. We show that essentially finite vector bundles coincide with those which become trivial after being pulled back by some proper and surjective morphism to X.

scholarly journals Barrelled Linearly Topologized Modules Over a Discrete Valuation Ring

2020 ◽  
Vol 12 (1) ◽  
pp. 69
Author(s):  
Dinamérico P. Pombo Jr ◽  
Patricia Couto G. Mauro
Keyword(s):  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Continuous Linear ◽  
Steinhaus Theorem

In this paper barrelled linearly topologized modules over an arbitrary discrete valuation ring are introduced. A general form of the Banach-Steinhaus theorem for continuous linear mappings on barrelled linearly topologized modules is established and some consequences of it are derived.

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