Hermitian K-theory for stable $\infty$-categories III: Grothendieck-Witt groups of rings - Archive ouverte HAL Access content directly
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Hermitian K-theory for stable $\infty$-categories III: Grothendieck-Witt groups of rings

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1
Baptiste Calmès
Emanuele Dotto
  • Function : Author
Yonatan Harpaz
Fabian Hebestreit
  • Function : Author
Markus Land
  • Function : Author
Kristian Moi
  • Function : Author
Denis Nardin
  • Function : Author
Thomas Nikolaus
  • Function : Author
Wolfgang Steimle
  • Function : Author

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.

Dates and versions

hal-02941257 , version 1 (16-09-2020)

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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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