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

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.
Document type :
Preprints, Working Papers, ...

https://hal-univ-artois.archives-ouvertes.fr/hal-02941257
Contributor : Baptiste Calmès Connect in order to contact the contributor
Submitted on : Wednesday, September 16, 2020 - 9:18:17 PM
Last modification on : Friday, January 21, 2022 - 3:30:50 AM

### Identifiers

• HAL Id : hal-02941257, version 1
• ARXIV : 2009.07225

### Citation

Baptiste Calmès, Emanuele Dotto, yonatan Harpaz, Fabian Hebestreit, Markus Land, et al.. Hermitian K-theory for stable $\infty$-categories III: Grothendieck-Witt groups of rings. 2020. ⟨hal-02941257⟩

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