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

To Andrew Ranicki

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 $\mathrm{K}$-theory and Ranicki’s original (non-periodic) symmetric $\mathrm{L}$-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 cohrent rings of global dimension $d$ is an equivalence in degrees $\le d+3$. As an important tool, we establish the hermitian analogue of Quillen’s localisation-dévissage sequence for Dedekind rings and use it to solve a conjecture of Berrick-Karoubi.

Authors

Baptiste Calmès

Université d'Artois, Laboratoire de Mathématiques de Lens (LML), Lens, France

Emanuele Dotto

University of Warwick, Mathematics Institute, Coventry, United Kingdom

Yonatan Harpaz

Université Paris 13, Institut Galilée, Villetaneuse, France

Fabian Hebestreit

RFWU Bonn, Mathematisches Institut, Bonn, Germany

Markus Land

Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark

Kristian Moi

KTH, Institutionen för Matematik, Stockholm, Sweden

Denis Nardin

Universität Regensburg, Mathematisches Institut, Regensburg, Germany

Thomas Nikolaus

WWU Münster, Mathematisches Institut, Münster, Germany

Wolfgang Steimle

Universität Augsburg, Institut für Mathematik, Augsburg, Germany