1
$\begingroup$

Let $R$ be a PID and $M$ a free, finite rank $R$-module with a perfect billinear form $\omega$ such that $\omega(v,v)=0$ for all $v \in M$. Does anyone know a reference for the fact that a symplectic basis exists? By symplectic basis, I mean a basis $x_1,\ldots x_n,y_1 \ldots y_n$ of $M$ with $\omega(x_i,x_j)=0$, $\omega(y_i,y_j)=0$, and $\omega(x_i,y_j)=\delta_{i,j}$.

$\endgroup$
3
$\begingroup$

See, for example, Max-Albert Knus, Quadratic and Hermitian forms over rings, Chapter I, Corollary 4.1.2. Or just mimic a proof in the case of fields (note, however, that "perfect" here should mean that the determinant of the Gram matrix is invertible, not just non zero).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.