Let $X$ a complex manifold, and $\mathcal O_X$ the sheaf of holomorphic functions. Oka Coherence Theorem states that $\mathcal O_X$ is coherent (as $\mathcal O_X$-module).
How to prove that also the sheaf of rings $\mathcal O_X[T]$, defined as the sheaf associated to the presheaf $\mathcal O_X\otimes_\mathbb Z \mathbb Z[T]$, is coherent (as $\mathcal O_X[T]$-module)?
More in general, if $(X,\mathcal A)$ is a (locally) ringed space with coherent structural sheaf, what are the conditions (on the space $X$, or on the sheaf $\mathcal A$) to require so that also the ringed space $(X,\mathcal A[T])$ has a coherent structural sheaf?
In general, using the Soublin counterexample (http://www.sciencedirect.com/science/article/pii/0021869370900505#), it should be easy to construct examples of ringed spaces which do not satisfy this property.
