A hermitian analog of a quadratic form theorem of Springer

Abstract

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.