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 $\mathbb{ F}_2[x]/(x^n)$.
Categories: math.KT math.RA