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The Inverse Galois Problem for Connected Algebraic Groups

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Abstract

We show that each connected group scheme of finite type over an arbitrary ground field is isomorphic to the component of the identity inside the automorphism group scheme of some projective, geometrically integral scheme. The main ingredients are embeddings into smooth group schemes, equivariant completions, blow-ups of orbit closures, Fitting ideals for Kähler differentials, and Blanchard’s Lemma.

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Acknowledgements

This research started when the first author visited the Heinrich Heine University Düsseldorf, and was continued during two visits of the second author at the University of Grenoble. We both thank the host institutions for hospitality. This research was also conducted in the framework of the research training group GRK 2240: Algebro-geometric Methods in Algebra, Arithmetic and Topology, which is funded by the Deutsche Forschungsgemeinschaft. Finally, we thank the two referees for their careful reading and helpful comments.

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Authors and Affiliations

  1. Institut Fourier, Université Grenoble Alpes, CS 40700, 38058, Grenoble Cedex 9, France

    Michel Brion

  2. Faculty of Mathematics and Natural Sciences, Mathematical Institute, Heinrich Heine University Düsseldorf, 40204, Düsseldorf, Germany

    Stefan Schröer

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  1. Michel Brion
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  2. Stefan Schröer
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Correspondence to Michel Brion.

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Brion, M., Schröer, S. The Inverse Galois Problem for Connected Algebraic Groups. Transformation Groups (2024). https://doi.org/10.1007/s00031-024-09865-0

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