Proposition 7.8 in Atiyah MacDonald

Asked Mar 21 '17 at 09:01

Active Mar 21 '17 at 09:01

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I am reading "Introduction to Commutative Algebra" by Atiyah and MacDonald, and I am stuck at a detail in Proposition 7.8:

Let $A\subset B\subset C$ be rings. Suppose that $A$ is Noetherian, that $C$ is finitely generated as $A$-algebra and that $C$ is either i) finitely generated as a $B$-module or ii) integral over $B$. Then $B$ is finitely generated as an $A$-algebra.

I understand everything in the proof except the first line: Condition i) and ii) are equivalent in this situation.

Why is this the case?

asked Mar 21 '17 at 09:01

user427787

1

Since $C$ is a finitely generated $B$-algebra. – user26857 Mar 21 '17 at 09:04
@user26857. I do not understand, could you please fill in some details? – user427787 Mar 21 '17 at 09:16
2

finitely generated as algebra + integral = finitely generated as a module. – MooS Mar 21 '17 at 09:21
@MooS, @ user26857. Of course, I understand now. Thank you. – user427787 Mar 21 '17 at 09:28

Proposition 7.8 in Atiyah MacDonald

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