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.