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.
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Acknowledgements
I would like to thank Uriya First and Stephen Scully for useful discussions and comments on earlier versions of this work, and the referee for comments and corrections.
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Gille, S. A hermitian analog of a quadratic form theorem of Springer. manuscripta math. 163, 125–163 (2020). https://doi.org/10.1007/s00229-019-01155-4
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DOI: https://doi.org/10.1007/s00229-019-01155-4