Serre wrote in his letter to Grothendieck(Oct. 25,1959) that valuation rings are coherent. How do you prove it?
Are valuation rings coherent?
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commutative-algebra
1 Answers
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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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0For 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. – 2012-11-20