# TOWERS OF TORSORS OVER A FIELD

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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https://hal.univ-cotedazur.fr/hal-01342094
Contributor : Marco Antei <>
Submitted on : Tuesday, July 5, 2016 - 2:04:57 PM
Last modification on : Tuesday, May 26, 2020 - 6:50:52 PM
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• HAL Id : hal-01342094, version 1

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Marco Antei, Indranil Biswas, Michel Emsalem. TOWERS OF TORSORS OVER A FIELD. 2016. ⟨hal-01342094⟩

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