are given by rows $\uni {d+1}a$ such that the $a_i$’s generate $A$. Such a row is called unimodular, and it is easily seen that there is an action of $GL_{d+1}(A)$ on the set $Um_{d+1}(A)$ of unimodular rows of rank $d+1$. A stably free module is free if and only if its associated unimodular row is in the $GL_{d+1}(A)$ orbit of $(1,0,\ldots,0)$. This obstruction is “computable” in the following sense: If $E_{d+1}(A)$ is the subgroup of $GL_{d+1}(A)$ generated by the elementary matrices, then there is a structure of abelian group on $Um_{d+1}(A)/E_{d+1}(A)$ ([@vdk Theorem 4.1]). If moreover $A$ is a smooth $k$-algebra of dimension $d\geq 3$ over a perfect field of characteristic different from $2$, this group can be seen as the cohomology group $H^d(A,G^{d+1})$, where $G^{d+1}$ is a suitable sheaf on $\spec A$ ([@Fa4 Theorem 4.9]). The induced action of $SL_{d+1}(A)$ on $Um_{d+1}(A)/E_{d+1}$ reads in this particular situation as a group homomorphism $SL_{d+1}(A)/E_{d+1}(A)\to Um_{d+1}(A)/E_{d+1}(A)$ defined by sending a matrix to its first row. The cokernel of this map is in bijection with $Um_{d+1}(A)/SL_{d+1}(A)$, which is therefore computable in some situations (see [@Fa4 Theorem 5.6] for some cases).

If the projective module $P$ is of rank $r<d$, then the situation is 