condition *locally free* is necessary for this equivalence.

Let $\omega:{\mathcal{L}}\to\Omega^q$ be a integrable $q$-form. Then, we can consider two maps, $$\xymatrix{
\bigwedge^{q-1} TX\otimes {\mathcal{L}}\ar[r]^<<<<<{i_{(\cdot)}\omega}
&\Omega^1_X\ar[r]^<<<<<{(\cdot)\wedge\omega}&\Omega_X^{q+1}\otimes{\mathcal{L}}^{-1}
}$$ The integrability condition on $\omega$ implies that this diagram is a complex and it is easy to check that its homology is supported over the points where $\omega$ is not decomposable. We define the sheaf associated to $\omega$, denoted ${\mathcal{E}}={\mathcal{E}}(\omega)$, as the kernel of $(\cdot)\wedge\omega$. By definition, ${\mathcal{E}}$ is a reflexive sheaf.

\[exampleCodimension2inA3\] Let $X=\mathbb{A}^3$ or, in the holomorphic case, a polydisk of dimension $3$. We take $v\in \Gamma(X,TX)$ a vector field, generic in the sense that in a coordinate system $(x_1,x_2,x_3)$ we can write $v=f_1\frac{\partial}{\partial x_1}-f_2\frac{\partial}{\partial x_2}+f_3\frac{\partial}{\partial x_3}$ with $f_1, f_2, f_3 \in k[x_1,x_2,x_3]$ and such that the ideal $(f_1, f_2,f_3)\subseteq k[x_1,x_2,x_3]$ is a complete intersection, that is, there are no nontrivial relations among the $f_i$’s.

The vector field $v$ generates a codimension $2$ foliation in $X$, this foliation is determined by a $2$-form $\omega$ such that $i_v\omega=0$. One such $\omega$ is given by $$\omega= f_3 dx_1\wedge dx_2 + f_2 dx_1\wedge dx_3 + f_1 dx_2\wedge dx_3.$$ It can be verified that this $\omega$ satisfies Plücker relations, is integrable, $i_v\omega=0$ and that $\Omega^2_X/(\omega)$ is 