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Enhanced Koszul properties in Galois cohomology

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Abstract

We prove that Galois cohomology satisfies several surprisingly strong versions of Koszul properties, under a well known conjecture, in the finitely generated case. In fact, these versions of Koszulity hold for all finitely generated maximal pro-p quotients of absolute Galois groups which are currently understood. We point out several of these unconditional results which follow from our work. We show how these enhanced versions are preserved under certain natural operations on algebras, generalising several results that were previously established only in the commutative case. The subject matter in this paper contains topics which are used in various branches of algebra and computer science.

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Notes

  1. The chosen and strongly defended by the second author, despite the doubts of the third author.

  2. The tag \(\heartsuit \) represents the authors’ love for this property.

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Acknowledgements

We are very grateful to Sunil Chebolu, Ronie Peterson Dario, Ido Efrat, David Eisenbud, Jochen Gärtner, Stefan Gille, Pierre Guillot, Chris Hall, Daniel Krashen, John Labute, Eliyahu Matzri, Danny Neftin, Jasmin Omanovic, Claudio Quadrelli, Andrew Schultz, John Tate, Ştefan Tohǎneanu, Adam Topaz, Thomas Weigel and Olivier Wittenberg for interesting discussions, enthusiasm, and encouragement concerning this paper and related topics. The authors are extremely grateful to the referee for her/his insightful report and for the very careful and thoughtful comments, suggestions and corrections. These suggestions and corrections have helped us to make our paper clearer and more readable. Ján Mináč gratefully acknowledges support from Natural Sciences and Engineering Research Council of Canada (NSERC) Grant R0370A01. Nguyễn Duy Tân is partially supported by the Vietnam Academy of Science and Technology under Grant Number CT0000.02/20-21.

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  1. University of Western Ontario, London, ON, Canada

    Ján Mináč, Marina Palaisti & Federico W. Pasini

  2. Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, 10307, Vietnam

    Nguyễn Duy Tân

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  1. Ján Mináč
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  4. Nguyễn Duy Tân
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Correspondence to Federico W. Pasini.

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Dedicated to David Eisenbud, with admiration and gratitude.

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Mináč, J., Palaisti, M., Pasini, F.W. et al. Enhanced Koszul properties in Galois cohomology. Res Math Sci 7, 10 (2020). https://doi.org/10.1007/s40687-020-00208-5

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