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Rationally Isomorphic Hermitian Forms and Torsors of Some Non-Reductive Groups
Let be a semilocal Dedekind domain. Under certain assumptions, we show
that two (not necessarily unimodular) hermitian forms over an -algebra with
involution, which are rationally ismorphic and have isomorphic semisimple
coradicals, are in fact isomorphic. The same result is also obtained for
quadratic forms equipped with an action of a finite group. The results have
cohomological restatements that resemble the Grothendieck--Serre conjecture,
except the group schemes involved are not reductive. We show that these group
schemes are closely related to group schemes arising in Bruhat--Tits theory.Comment: 27 pages. Changes from previous version: Section 5 was split into two
sections, several proofs have been simplified, other mild modification
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