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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.

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1 Answer 1

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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).

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