A presentation of the symmetric Grothendieck-Witt group of local rings over $\mathbb{F}_2$
Marcus Nicolas
TL;DR
This paper addresses the problem of giving explicit abelian-group presentations for the symmetric Grothendieck–Witt group $\mathrm{GW}^{\mathrm{s}}(R)$ and the symmetric Witt group $\mathrm{W}^{\mathrm{s}}(R)$ for commutative local rings $R$ with residue field $\mathbb{F}_2$. Building on the general presentation of $\mathrm{GW}^{\mathrm{s}}(R)$ via diagonal congruences from KRW72, it analyzes the obstruction to lifting diagonal congruences through the residue field and reduces the kernel to two families of relations (even and odd) arising from diagonal 4×4 congruences. The main result shows that $\mathrm{GW}^{\mathrm{s}}(R)$ is presented by generators $\langle a\rangle$ with $a\in R^{\times}/R^{\times 2}$ subject to these two families of relations, enabling explicit calculations of $\mathrm{GW}^{\mathrm{s}}(R)$ and $\mathrm{W}^{\mathrm{s}}(R)$ for rings such as $\mathbb{Z}/2^n\mathbb{Z}$ and $\mathbb{F}_2[x]/(x^n)$. The applications yield precise group and ring structures, describe symmetrisation maps and Arf-type invariants, and identify towers and limits with 2-adic and formal power-series analogues, thereby providing a complete computational framework in characteristic 2 settings.
Abstract
Let $R$ be a commutative local ring. We provide an explicit presentation of the symmetric Grothendieck-Witt ring $\mathrm{GW}^{\mathrm{s}}(R)$ of $R$ as an abelian group when $R$ has residue field $\mathbb{F}_2$. This completes a recent work by Rogers and Schlichting, where an explicit presentation of $\mathrm{GW}^{\mathrm{s}}(R)$ is given when the residue field is different from $\mathbb{F}_2$. We then use this result to compute the symmetric Grothendieck-Witt rings for the sequences of local rings $\mathbb{Z}/2^n\mathbb{Z}$ and $\mathbb{F}_2[x]/(x^n)$.