Suppose that $f:P\to Q$ is an isomorphism of (rank $2$) projective modules. Then $\det f$ induces an isomorphism $\Ko A{\det P}\to \Ko A{\det Q}$ under which $e(P)$ is sent to $e(Q)$. The Euler class satisfies good functorial properties, as shown in [@FS §2.4, §2.5]. In particular,

\[vanishing\] Suppose that $P\simeq (\det P)\oplus A$. Then $e(P)=0$.

If $A$ is of dimension $2$, then the proposition is the easy part of a stronger result ([@FS Proposition 23]):

Let $A$ be a ring of dimension $2$. Then $e(P)=0$ if and only if $P\simeq (\det P)\oplus A$.

If $A$ is of dimension $3$, then the above result is no longer true, and even for a stably free module of rank $2$ we need another obstruction to detect whether it is free or not. This is the object of the next section.

Unimodular rows under the symplectic group {#unimodular}
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Unimodular rows
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A unimodular row of length $n$ is a surjective homomorphism $A^n\to A$. Equivalently, a unimodular row is represented by a set $\uni na$ of elements of $A$ which generate the ideal $A$. The group $GL_n(A)$ acts on $Um_n(A)$ by composition, and therefore any subgroup of $GL_n(A)$ also acts. Traditionally, one is interested 