A Hermitian analog of a quadratic form theorem of Springer
Let \(F\) be a complete discretely valued field of characteristic not 2 with residue field \(\overline{F}\). \textit{T. A. Springer} [Nederl. Akad. Wet., Proc., Ser. A 58, 352--362 (1956; Zbl 0067.27605)] showed that the Witt group of symmetric bilinear spaces over \(F\) decomposes as \(W(F) \cong W(\overline{F}) \oplus W(\overline{F})\). A corollary of this computation is the existence of second residue maps \(W(K) \rightarrow W(k)\), where \(K\) is the fraction field of a discrete valuation ring \(R\) with residue field \(k\), whose characteristic is not two. This map, denoted \(\partial_\pi\) here, depends on the choice of an uniformizer \(\pi\), i.e., generator of the maximal ideal \(\mathfrak{m}\) of \(R\), and fits into an exact sequence as the third map \[0 \rightarrow W(R) \rightarrow W(K)\rightarrow W(k) \rightarrow 0.\] The goal of this paper is to prove the analogous results for Witt groups of algebras with involution: Theorem B. Let \(R\) be a complete discrete valuation ring with fraction field \(K\) and residue field \(k\), and \((A, \tau)\) an \(R\)-Azumaya algebra with involution of the first kind. Then \(W_\varepsilon(A_K , \tau_K ) \cong W_\varepsilon(A_k, \tau_K ) \oplus W_\varepsilon(A_k, \tau_k )\) for any \(\varepsilon \in \{\pm 1\}\). Theorem A. Let \(R\) be a semilocal Dedekind domain containing \(\frac{1}{2}\) with fraction field \(K\) and \((A, \tau)\) an \(R\)-Azumaya algebra with involution of first or second kind. We assume that \(R\) is Galois of degree 2 over the fix ring of \(\tau|_R\) if \(\tau\) is not of the first kind. Let \(\max R \subset \operatorname{Spec} R\) be the set of maximal ideals of \(R\) and \(X_\tau^{(1)} := \{\mathfrak{m} \in \max R \ | \ \tau(\mathfrak{m}) = \mathfrak{m}\} = \{\mathfrak{m}_1,\dots ,\mathfrak{m}_\ell\}\). (\(\ell = 0\) is possible). We set \(k_i := R/\mathfrak{m}_i\) for \(i = 1,\dots, \ell\), and denote by \(\tau_K\) (respectively, \(\tau_{k_i}\)) the induced involutions on \(A_K:= K\otimes_R A\) (and \(A_{k_i}:= k_i\otimes_R A\), \(1 \leq i \leq \ell\)). [Note that since \(R\) is Galois over the fix ring of \(\tau|_R\) these involutions have the same kind as \(\tau\).] Then there exists an exact sequence of \(\varepsilon\)-hermitian Witt groups (\(\varepsilon=\pm 1\)), \[0\rightarrow W(A,\tau) \rightarrow W(A_K,\tau_K)\rightarrow \bigoplus_{i=1}^\ell W(A_{k_i},\tau_{k_i}) \rightarrow 0,\] where the ``second residue maps'' \(d^{\varepsilon,\pi_i}_{A_{\mathfrak{m}_i},\tau_{\mathfrak{m}_i}}\) (by which the third map in the sequence is given) depend on the choice of a local uniformizer \(\pi_i\) at the maximal ideal \(\mathfrak{m}_i\). In the last section, as a corollary of Theorem A, the author concludes that if \((A, \tau)\) is an Azumaya algebra with involution of the first or second kind over a semilocal regular domain \(R\) of dimension 2 with fraction field \(K\) then the natural homomorphism of \(\varepsilon\)-hermitian Witt groups \(W_\varepsilon(A, \tau) \rightarrow W_\varepsilon(A_K , \tau_K )\) is a monomorphism onto the unramified \(\varepsilon\)-hermitian Witt group of \((A,\tau)\).
scientific article; zbMATH DE number 7233346
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