  Given a generically surjective holomorphic vector bundle morphism $f\colon
E\to Q$, $E$ and $Q$ Hermitian bundles, we construct a current $R^f$ with
values in $\Hom(Q,H)$, where $H$ is a certain derived bundle, and with support
on the set $Z$ where $f$ is not surjective. The main property is that if $\phi$
is a holomorphic section of $Q$, and $R^f\phi=0$, then locally $f\psi=\phi$ has
a holomorphic solution $\psi$. In the generic case also the converse holds.
This gives a generalization of the corresponding theorem for a complete
intersection, due to Dickenstein-Sessa and Passare. We also present results for
polynomial mappings, related to M Noether's theorem and the effective
Nullstellensatz. The construction of the current is based on a generalization
of the Koszul complex. By means of this complex one can also obtain new global
estimates of solutions to $f\psi=\phi$, and as an example we give new results
related to the $H^p$-corona problem.
