Is a coherent ring direct limit of Noetherian rings?

Asked Sep 15 '17 at 06:50

Active Sep 15 '17 at 08:52

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A commutative ring is coherent if every finitely generated ideal is finite presented. Every Noetherian ring is coherent.

I know polynomial rings with infinite variables is coherent but not Noetherian. And it is direct limit of Noetherian rings in the category of ring. My question: is the converse true? That is, every coherent ring is a direct limit of Noetherian rings?
Is there any relation between coherent rings and coherent sheafs?

Thank you in advance!

edited Sep 15 '17 at 08:52

user26857

asked Sep 15 '17 at 06:50

Jian

Adapt this: https://math.stackexchange.com/questions/1680007/polynomial-ring-in-infinitely-many-variables-over-a-noetherian-ring-is-coherent – Mariano Suárez-Álvarez Sep 15 '17 at 06:55
@MarianoSuárez-Álvarez when you answer my question,than I prove this by using extension and contraction of ideals.I want to know the converse.Thank your link! – Jian Sep 15 '17 at 07:04
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See this answer. – user26857 Sep 15 '17 at 08:45

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