Rationally isomorphic quadratic objects

Let R be a discrete valuation ring with fraction field F. Two algebraic objects (say, quadratic forms) defined over R are said to be rationally isomorphic if they become isomorphic after extending scalars to F. In the case of unimodular quadratic forms, it is a classical result that rational isomorphism is equivalent to isomorphism. This has been recently extended to "almost umimodular" forms by Auel, Parimala and Suresh. I will present further generalizations to hermitian forms over (certain) involutary R-algebras and quadratic spaces equipped with a group action ("G-forms"). The results can be regarded as versions of the Grothendieck-Serre conjecture for certain non-reductive groups.

(Joint work with Eva Bayer-Fluckiger.)