Patching And Weak Approximation In Isometry Groups

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 ( not 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.