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In "stack project", there is a lemma on finite locally free morphisms, saying that a finite locally free morphism of schemes is equivalent to a morphism which is finite, flat, and locally of finite presentation.

For the proof, they refer to the commutative algebra fact that a module is finite locally free iff it is flat and finitely presented.

In order to complete the proof, I have the following gap:

Let $A \to B$ be a morphism of rings. Then $B$ can either be viewed as an $A$-algbra or an $A$-module. Is it true that the following two statements are equivalent?

  1. $B$ is a flat $A$-module and is finitely presented as an $A$-module.
  2. $B$ is a finitely generated flat $A$-module, and is finitely presented as an $A$-algebra.
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    $\begingroup$ Yes, this is true, a finite and finitely presented algebra is finitely presented as a module. This is somewhere in EGA. $\endgroup$
    – Angelo
    Commented Jan 1, 2013 at 8:41
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    $\begingroup$ EGA IV$_1$, 1.4.7. $\endgroup$
    – user30180
    Commented Jan 1, 2013 at 8:49
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    $\begingroup$ @ayanta: why don't you make this an answer, so the OP can accept it. Otherwise this question might pop up to the frontpage as unanswered at some point(s) in the future. Either that, or the OP could close the question, but I like the first solution better $\endgroup$
    – David White
    Commented Jan 1, 2013 at 18:24
  • $\begingroup$ I like your 2nd solution better, but I have posted the answer you requested. $\endgroup$
    – user30180
    Commented Jan 1, 2013 at 21:28

2 Answers 2

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EGA IV$_1$, 1.4.7.

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It appears that in the meantime, full proofs have been added to the Stacks Project.

Tag 0564. Let $R \to S$ be a finite and finitely presented ring map. Let $M$ be an $S$-module. Then $M$ is finitely presented as an $R$-module if and only if $M$ is finitely presented as an $S$-module.

In particular, $S$ is finitely presented as an $R$-module if $S$ is finitely generated as an $R$-module and finitely presented as an $R$-algebra.

Tag 058R. Let $M$ be an $R$-module. Then $M$ is finite projective if and only if $M$ is finitely presented and flat.

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    $\begingroup$ Fun fact: according to stacks.math.columbia.edu/tag/0564/history, it was already in there in 2010 :). $\endgroup$
    – pbelmans
    Commented May 22, 2017 at 20:15
  • $\begingroup$ @pbelmans Yes, it seems that I was reading a very old vesion.... $\endgroup$
    – Xiaobo Zhuang
    Commented Feb 19, 2020 at 1:12

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