某些非约化群的有理同构Hermitian型和Torsors
Rationally Isomorphic Hermitian Forms and Torsors of Some Non-Reductive Groups
摘要：
设$R$是半局部戴德金整环.在一定的假设下，我们证明了具有对合的$R$ -代数上的两个（不一定是幺模的）厄米特形式，它们是有理同构的，并且具有同构的半单余根，它们实际上是同构的.对于具有有限群作用的二次型也得到了同样的结果。结果有cohomological重述，类似于Grothendieck-塞尔猜想，除了所涉及的组计划是不还原的。我们表明，这些组计划是密切相关的组计划中产生的Bruhat-山雀理论。
英文摘要：
Let $R$ be a semilocal Dedekind domain. Under certain assumptions, we show that two (not necessarily unimodular) hermitian forms over an $R$-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.
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