Hermitian K-theory via oriented Gorenstein algebras

Marc Hoyois; Joachim Jelisiejew; Denis Nardin; Maria Yakerson

doi:10.1515/crelle-2022-0063

Published by De Gruyter October 27, 2022

Hermitian K-theory via oriented Gorenstein algebras

Marc Hoyois , Joachim Jelisiejew , Denis Nardin and Maria Yakerson

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2022-0063

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Abstract

We show that the hermitian K-theory space of a commutative ring R can be identified, up to 𝐀 1 -homotopy, with the group completion of the groupoid of oriented finite Gorenstein R-algebras, i.e., finite locally free R-algebras with trivialized dualizing sheaf. We deduce that hermitian K-theory is universal among generalized motivic cohomology theories with transfers along oriented finite Gorenstein morphisms. As an application, we obtain a Hilbert scheme model for hermitian K-theory as a motivic space. We also give an application to computational complexity: we prove that 1-generic minimal border rank tensors degenerate to the big Coppersmith–Winograd tensor.

Funding statement: Marc Hoyois, Denis Nardin, and Maria Yakerson were partially supported by SFB 1085 “Higher invariants”. Joachim Jelisiejew was supported by NCN grant 2017/26/D/ST1/00913 and by the START fellowship of the Foundation for Polish Science.

Acknowledgements

We are thankful to Tom Bachmann, Joseph M. Landsberg, Rahul Pandharipande and Burt Totaro for helpful discussions. We would like to thank SFB 1085 “Higher invariants” and Regensburg University for its hospitality. Yakerson was supported by a Hermann-Weyl-Instructorship and is grateful to the Institute of Mathematical Research (FIM) and to ETH Zürich for providing perfect working conditions.

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Received: 2021-07-22

Revised: 2022-09-06

Published Online: 2022-10-27

Published in Print: 2022-12-01

Hermitian K-theory via oriented Gorenstein algebras

Abstract

Acknowledgements

References

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