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The following is Exercise 2.4, in Chapter 1 of Liu, Algebraic Geometry and Arithmetic Curves:

Let $I$ be a finitely generated ideal of $A$:

  1. $A/I$ is flat.

  2. $I^2 = I$.

  3. $I = (e)$ where $e^2=e$.

I can show that $2\iff 3$ and that $1 \implies 2$, and I remember proving the other way before but cannot recall it now. That is, I would like to show that $A/I$ is flat assuming that it is principal and generated by an idempotent.

user26857
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Asvin
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  • Related: http://math.stackexchange.com/questions/487214/is-r-nr-a-faithfully-flat-r-module – user26857 May 21 '16 at 16:56
  • Here's another question about the same problem, but the answers focus more on other implications: https://math.stackexchange.com/q/3145745/326036 – Anakhand Nov 02 '23 at 14:19

1 Answers1

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I am sure this is proved somewhere on here, but i could not find the link. But here it is: $A=(e)\times (1-e)$, so $(1-e)\cong A/I$ is projective.