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

Abstract

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.