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arXiv · 2402.15136

A motivic spectrum representing hermitian K-theory

Baptiste Calmès·Yonatan Harpaz·Denis Nardin
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Abstract

We establish fundamental motivic results about hermitian K-theory without assuming that 2 is invertible on the base scheme. In particular, we prove that both quadratic and symmetric Grothendieck-Witt theory satisfy Nisnevich descent, and that symmetric Grothendieck-Witt theory further satisfies a projective bundle formula, as well as d\'evissage and A^1-invariance over a regular Noetherian base of finite Krull dimension. We use this to show that over a regular Noetherian base, symmetric Grothendieck-Witt theory is represented by a motivic E-infinity-ring spectrum, which we then show is an absolutely pure spectrum, answering a question of D\'eglise. As with algebraic K-theory, we show that over a general base, one can also construct a hermitian K-theory motivic spectrum, representing this time a suitable homotopy invariant and Karoubi-localising version of Grothendieck-Witt theory.

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Baptiste Calmès, Yonatan Harpaz, Denis Nardin. 2024-02-23. A motivic spectrum representing hermitian K-theory. https://arxiv.org/abs/2402.15136

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