Is every co-closed form also locally co-exact? That is for each $k$-form $\omega$ such that $\delta \omega = 0$ there exists $(k-1)$-form $\eta$ for which locally $\omega = \delta \eta$.
My current books on differential geometry are have nothing on the subject, so if this is indeed true I would also appreciate a book(an-article) I could formally reference to.
Just use the formula for the codifferential in terms of the differential and the Hodge dual: $\delta = \pm \star d \star$ where the sign depends on $k$, the dimension of the space, etc.
The Hodge star is an isomorphism, so $\delta \omega = 0$ if and only if $d \star \omega = 0$. Thus by the Poincaré Lemma we know $\star \omega = d\eta$ and thus $\omega = \pm\star d \eta = \pm \delta (\star\eta)$. (I was sloppy with the sign throughout - you don't really need to keep track of it, because at worst you just change the sign of $\star \eta$.)