On inner product spaces over Dedekind domains of characteristic two

Abstract:

Suppose $D$ is a Dedekind domain of characteristic 2 and $(M,\varphi )$ is an inner product space, i.e. $M$ is a finitely generated projective $D$ module supplied with a nonsingular symmetric bilinear form $\varphi$. It is shown that $(M,\varphi )$ is determined up to isometry by the extension of $\varphi$ to $F{ \otimes _D}M$, where $F$ is the quotient field of $D$, and the value module ${\mathcal Q}(M)$ of all $\varphi (m,m)$ for $m$ in $M$. In particular, a hyperbolic space ${\mathbf {H}}(M)$ is completely determined by the rank of the finitely generated projective module $M$. As consequences, genera coincide with isometry classes, and if ${N_1}$ and ${N_2}$ are isometric nonsingular submodules of $(M,\varphi )$ such that ${\mathcal Q}({N_1}^ \bot ) = {\mathcal Q}({N_2}^ \bot )$, then ${N_1}^ \bot$ and ${N_2}^ \bot$ are isometric. Also, given an $F$ inner product space $(V,\varphi )$ and a ${D^{(2)}}$ submodule $P$ of $D$, a necessary and sufficient condition is given for the existence of a $D$ inner product space $(M,\Psi )$ such that $(FM,\Psi ) \cong (V,\varphi )$ and ${\mathcal Q}(M) = P$.