A presentation of the symmetric Grothendieck–Witt group of local rings over 𝔽2

Let $R$ be a commutative local ring. We provide an explicit presentation of the symmetric Grothendieck–Witt ring ${GW}^{s} (R)$ of $R$ as an abelian group when $R$ has residue field $𝔽_{2}$ . This completes the work of Rogers and Schlichting (Math. Z. 307:2 (2024), art. id. 41), where an explicit presentation of ${GW}^{s} (R)$ is given when the residue field is different from $𝔽_{2}$ . We then use this result to compute the symmetric Grothendieck–Witt rings for the sequences of local rings $ℤ ∕ 2^{n} ℤ$ and $𝔽_{2} [x] ∕ (x^{n})$ .

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Keywords

Grothendieck–Witt theory, presentation, local ring, cyclic group, truncated polynomial, formal power series

Mathematical Subject Classification

Primary: 19G12

Milestones

Received: 25 May 2025

Revised: 27 August 2025

Accepted: 22 September 2025

Published: 5 November 2025

Authors

Marcus Nicolas
Ludwig Maximilian University of Munich Munich Germany

Vol. 10, No. 4, 2025

Marcus Nicolas

Abstract

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Keywords

Mathematical Subject Classification

Milestones

Authors