Finiteness theorems for the shifted Witt and higher Grothendieck-Witt groups of arithmetic schemes

摘要

For smooth varieties over finite fields, we prove that the shifted (aka derived) Witt groups of surfaces are finite and the higher Grothendieck-Witt groups (aka Hermitian K-theory) of curves are finitely generated. For more general arithmetic schemes, we give conditional results, for example, finite generation of the motivic cohomology groups implies finite generation of the Grothendieck-Witt groups.