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arXiv · 2009.07225

Hermitian K-theory for stable $\infty$-categories III: Grothendieck-Witt groups of rings

Baptiste Calmès·Emanuele Dotto·Yonatan Harpaz·Fabian Hebestreit·Markus Land·Kristian Moi·Denis Nardin·Thomas Nikolaus·Wolfgang Steimle
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Abstract

We establish a fibre sequence relating the classical Grothendieck-Witt theory of a ring $R$ to the homotopy $\mathrm{C}_2$-orbits of its K-theory and Ranicki's original (non-periodic) symmetric L-theory. We use this fibre sequence to remove the assumption that 2 is a unit in $R$ from various results about Grothendieck-Witt groups. For instance, we solve the homotopy limit problem for Dedekind rings whose fraction field is a number field, calculate the various flavours of Grothendieck-Witt groups of $\mathbb{Z}$, show that the Grothendieck-Witt groups of rings of integers in number fields are finitely generated, and that the comparison map from quadratic to symmetric Grothendieck-Witt theory of Noetherian rings of global dimension $d$ is an equivalence in degrees $\geq d+3$. As an important tool, we establish the hermitian analogue of Quillen's localisation-d\'evissage sequence for Dedekind rings and use it to solve a conjecture of Berrick-Karoubi.

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Baptiste Calmès, Emanuele Dotto, Yonatan Harpaz, Fabian Hebestreit, Markus Land, Kristian Moi, Denis Nardin, Thomas Nikolaus, Wolfgang Steimle. 2020-09-15. Hermitian K-theory for stable $\infty$-categories III: Grothendieck-Witt groups of rings. https://arxiv.org/abs/2009.07225

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