The Stacks project

Proof. We can see that the integer $d(M, \varphi (M))$ does not depend on the lattice $M$ as follows. Suppose that $M'$ is a second such lattice. Then we see that

Since $\varphi $ is an isomorphism we see that $d(\varphi (M'), \varphi (M)) = d(M', M) = -d(M, M')$, and hence $d(M, \varphi (M)) = d(M', \varphi (M'))$. Moreover, both sides of the equation (of the lemma) are additive in $\varphi $, i.e.,

and also

by the independence shown above. Hence it suffices to prove the lemma for generators of $\text{GL}(V)$. Choose an isomorphism $K^{\oplus n} \cong V$. Then $\text{GL}(V) = \text{GL}_ n(K)$ is generated by elementary matrices $E$. The result is clear for $E$ equal to the identity matrix. If $E = E_{ij}(\lambda )$ with $i \not= j$, $\lambda \in K$, $\lambda \not= 0$, for example

then with respect to a different basis we get $E_{12}(1)$. The result is clear for $E = E_{12}(1)$ by taking as lattice $R^{\oplus n} \subset K^{\oplus n}$. Finally, if $E = E_ i(a)$, with $a \in K^*$ for example

then $E_1(a)(R^{\oplus b}) = aR \oplus R^{\oplus n - 1}$ and it is clear that $d(R^{\oplus n}, aR \oplus R^{\oplus n - 1}) = \text{ord}_ R(a)$ as desired. $\square$