I was wondering if for an arbitrary valuation ring $R$ and an element $\pi$ in the maximal ideal of $R$, the rings


1. $R[x\_1,\ldots,x\_n]$;
2. projective limit of $R/(\pi^l)[x\_1,\ldots,x\_n]$, probably also denoted as $R \langle x\_1,\ldots,x\_n \rangle$ in the case when $R$ is a rank 1 valuation ring and;
3. $R/(\pi)[x\_1,\ldots,x\_n]$


are coherent or not?


The obvious implication being that either (1) or (2) would imply (3), and I think in the rank 1 valuation case all three rings considered above are indeed coherent. But I really don't know anything about this question in general. Any comment would be helpful.



Is polynomial ring of n variables with coefficients in an arbitrary valuation ring coherent?
Concerning question (1):



> 
> Polynomial algebras in finitely many indeterminates over valuation domains are coherent.
> 
> 
> 


This is Theorem 7.3.3 in S. Glaz, *Commutative coherent rings,* Springer LNM 1371 (1989).


