Abstract

Let $R$ be a commutative local ring. We provide an explicit presentation of the symmetric Grothendieck–Witt ring ${\mathrm{GW}<!--FUNCTION\; APPLICATION-->}^{s<!--FUNCTION\; APPLICATION-->}(R)$ of $R$ as an abelian group when $R$ has residue field ${\mathbb{𝔽}}_2$. This completes the work of Rogers and Schlichting (Math. Z. 307:2 (2024), art. id. 41), where an explicit presentation of ${\mathrm{GW}<!--FUNCTION\; APPLICATION-->}^{s<!--FUNCTION\; APPLICATION-->}(R)$ 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 ${\mathbb{𝔽}}_2[x]∕(x^n)$.

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