Isometry群的修补与弱逼近
Patching and Weak Approximation in Isometry Groups
摘要：
设$R$是半局部主理想整环.在$R$上的两个代数对象中，标量扩张是有意义的（例如二次空间），如果它们在将标量扩张到$R$及其分数域的所有完备化之后变得同构，则称它们为同亏格。我们证明了（不一定交换的）$R$ -阶上的幺模二次空间的亏格中的同构类的个数总是$2$的有限幂，并且在进一步的假设下，例如阶是遗传的，这个个数是$1$。对于相关的对象，例如半双线性型的系统，也得到了同样的结果。证明中的一个关键成分是等距群的弱逼近定理，它在任何（拓扑）基域上都有效，甚至在半局部基环上也有效。附录部分证明了在较弱的假设下，具有对合的$R$ -阶上的二次空间的等距群可以看作光滑的仿射群格式。
英文摘要：
Let $R$ be a semilocal principal ideal domain. Two algebraic objects over $R$ in which scalar extension makes sense (e.g. quadratic spaces) are said to be of the same genus if they become isomorphic after extending scalars to all completions of $R$ and its fraction field. We prove that the number of isomorphism classes in the genus of unimodular quadratic spaces over (non necessarily commutative) $R$-orders is always a finite power of $2$, and under further assumptions, e.g. that the order is hereditary, this number is $1$. The same result is also shown for related objects, e.g. systems of sesquilinear forms. A key ingredient in the proof is a weak approximation theorem for groups of isometries, which is valid over any (topological) base field, and even over semilocal base rings. The appendix proves that the isometry group of a quadratic space over an $R$-order with involution can be regarded as a smooth affine group scheme under mild assumptions.
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