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

TL;DR

This paper develops a new fibre sequence linking Grothendieck-Witt theory and K-theory for rings, removing the 2 invertibility assumption, and applies it to compute and analyze Grothendieck-Witt groups in number theory.

Contribution

It introduces a fibre sequence connecting Grothendieck-Witt and K-theory, enabling new results in algebraic K-theory and quadratic forms without the 2 invertibility assumption.

Findings

Solved the homotopy limit problem for Dedekind rings with number field fraction fields.

Calculated Grothendieck-Witt groups of integers and rings of integers in number fields.

Established the hermitian analogue of Quillen's localisation sequence for Dedekind 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.