Towers of torsors over a field

2016/06/28 by Marco Antei, Indranil Biswas, Antei, Marco +3
Mathematics · #14F35 #14H30 #14L15 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1606.08671

openalex publication_date 2016/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a projective, connected and smooth scheme defined over an algebraically closed field k. In this paper we prove that a tower of finite torsors (i.e., under the action of finite k-group schemes) can be dominated by a single finite torsor. Let G be any finite k-group scheme and Y any G--torsor over X pointed in y ∈ Y(k); we define over Y, which may not be reduced, in a very natural way the categories of Nori-semistable and essentially finite vector bundles. These categories are proved to be Tannakian. Their Galois k-group schemes π^S(Y, y) and π^N(Y, y), respectively, thus generalize the S--fundamental and the Nori fundamental group schemes. The latter still classifies all the finite torsors over Y, pointed over y. We also prove that they fit in short exact sequences involving π^S(X, x) and π^N(X, x) respectively, where x is the image of y.

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