Exterior multiplication with singularities: a Saito's theorem on vector bundles

Let $E$ be a vector bundle over a suitable differential manifold $M$ and let $\wedge^p E$ denote $p$-exterior product of $E$. Given sections $\omega_1,\dots,\omega_k$ of $E$ and a section $\eta$ of $\wedge^p E$, we consider the problem if $\eta$ can be written in the form $$\eta=\sum \omega_i\wedge\gamma_i,$$ where $\gamma_i$ are sections of $\wedge^{p-1}E$. An obvious necessary condition $\Omega\wedge\eta=0$, where $\Omega=\omega_1\wedge\cdots\wedge\omega_k$, has to be supplemented with a condition that the form $\Omega$ has sufficiently regular singularities at points where $\Omega(x)=0$. Such a local condition is suggested by an algebraic theorem of K. Saito and is given in terms of the depth of the ideal defined by coefficients of $\Omega$. Working in the smooth, real analytic and holomorphic (with $M$ Stein manifold) categories, we show that the condition is sufficient for the above property to hold. Moreover, in the smooth category it is sufficient for existence of a continuous right inverse to the operator defined by $(\gamma_1,\dots,\gamma_k)\mapsto\sum \omega_i\wedge\gamma_i$. All these results are also proven in the case where $E$ is a bundle over a suitable closed subset of $M$.