Grothendieck-Witt groups of henselian valuation rings (Q6566575)

Let \((R, \mathfrak{m})\) be a local pair. A cohomology functor \(\mathcal{F}\) is said to be \textit{locally constant} at a local pair \((R, \mathfrak{m})\) if the natural map \(R \to R/\mathfrak{m}\) induces an isomorphism \(\mathcal{F}(R)\cong \mathcal{F}(R/\mathfrak{m}).\)\N\NA result of Suslin says that for a henselian pair \((R, \mathfrak{m})\), the higher \(K\)-theory with finite coefficient is locally constant (see [\textit{A. A. Suslin}, J. Pure Appl. Algebra 34, 301--318 (1984; Zbl 0548.12009)]). In this article, the author shows that the symplectic \(K\)-theory with finite coefficient is locally constant for a henselian pair (see Theorem 3.5). The author also shows that the higher Grothendieck-Witt groups with finite coefficient are locally constant for a henselian pair (see Theorem 4.4).