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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.

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  • $\begingroup$ The notion of stably coherent ring might be relevant for this - see, e.g., Section 7.3 in S. Glaz, Commutative coherent rings, Springer LNM 1371. $\endgroup$ Feb 5, 2015 at 20:48

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