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Motivated by an étale cohomology calculation I am going to do, here is a question that should have positive and not too hard answer.

Let $A$ be a ring, $A_0$ a ring such that $A_0$ is equipped with an endomorphism $f : A_0\to A_0$, and such that $A$ is the direct limit:

$$A = \varinjlim_{f^n : A_0\to A_0, n\ge 0} A_0.$$

Let $I\subset A$ be an ideal. Does there exist an ideal $I_0\subset A_0$ such that $I = \varinjlim f^{n}(I_0)$?

My guess is $$I_0 := \{x\in A_0\mid f^n(x)\in I\ \text{for some}\ n\}.$$

Improve this question
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    $\begingroup$ There is no reason to believe that such $\varinjlim f^n(I_0)$ is even an ideal of $A$. Note that, in particular, this will fail for $I=A$ (and $I_0=A_0$) whenever $f$ is not onto. Shouldn't the appropriate condition rather be something like "There is $I_0 \subseteq A_0$ invariant under $f$ with $I=\varinjlim_{f^n:I_0 \rightarrow I_0}I_0$"? $\endgroup$
    – Pavel Čoupek
    Commented Apr 15, 2018 at 4:08
  • $\begingroup$ How is this related to etale cohomology? $\endgroup$
    – user84144
    Commented Apr 20, 2018 at 21:25

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