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Serre wrote in his letter to Grothendieck(Oct. 25,1959) that valuation rings are coherent. How do you prove it?

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up vote 5 down vote accepted

The valuation takes values in a totally ordered group. So any finitely generated ideal is principal (take one of the generators with the smallest valuation), hence finitely presented.

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A principal ideal in a domain is free of rank 1, so finitely generated. – user18119 Nov 20 '12 at 12:44
Thanks. I had forgotten that finitely generated free(or more generally projective) modules are finitely presented. – Makoto Kato Nov 20 '12 at 13:54
For readers who may not know the following fact. Let $L_1, L_2$ be finitely generareted free modules over a ring. Let $f\colon L_1 \rightarrow L_2$ be an epimorphism. Since $L_2$ is free, the exact sequence $0 \rightarrow Ker(f) \rightarrow L_1 \rightarrow L_2 \rightarrow 0$ splits. Hence $Ker(f)$ is finitely generated. – Makoto Kato Nov 20 '12 at 14:04

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