We show that the second residue map for hermitian Witt groups of an Azumaya algebra A with involution $$\tau $$ of first- or second kind over a semilocal Dedekind domain R is surjective. This proves a generalization to hermitian Witt groups of an exact sequence for Witt groups of quadratic forms due to Springer. If R is a complete discrete valuation ring and $$\tau $$ is of the first kind we show that our short exact sequence of hermitian Witt groups is split. As a corollary we prove a purity theorem for hermitian Witt groups of Azumaya algebras with involutions over a regular semilocal domain of dimension two.

中文翻译：

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Springer 二次型定理的厄密模拟

我们表明，在半局部 Dedekind 域 R 上具有第一类或第二类对合 $$\tau $$ 的 Azumaya 代数 A 的厄密 Witt 群的第二个残差图是满射的。这证明了对 Springer 二次型 Witt 群的精确序列的厄密 Witt 群的推广。如果 R 是一个完全离散的估值环并且 $$\tau $$ 是第一种，我们证明我们的厄密维特群的短精确序列是分裂的。作为推论，我们证明了 Azumaya 代数 Hermitian Witt 群的纯度定理，该群在二维的规则半局域域上对合。