Lemma 15.93.1. Let $A$ be a ring. Let $f \in A$. If there exists an integer $c \geq 1$ such that $A[f^ c] = A[f^{c + 1}] = A[f^{c + 2}] = \ldots$ (for example if $A$ is Noetherian), then for all $n \geq 1$ there exist maps

$(A \xrightarrow {f^ n} A) \longrightarrow A/(f^ n), \quad \text{and}\quad A/(f^{n + c}) \longrightarrow (A \xrightarrow {f^ n} A)$

in $D(A)$ inducing an isomorphism of the pro-objects $\{ A/(f^ n)\}$ and $\{ (f^ n : A \to A)\}$ in $D(A)$.

Proof. The first displayed arrow is obvious. We can define the second arrow of the lemma by the diagram

$\xymatrix{ A/A[f^ c] \ar[r]_-{f^{n + c}} \ar[d]_{f^ c} & A \ar[d]^1 \\ A \ar[r]^{f^ n} & A }$

Since the top horizontal arrow is injective the complex in the top row is quasi-isomorphic to $A/f^{n + c}A$. We omit the calculation of compositions needed to show the statement on pro objects. $\square$

Comment #230 by on

There is a typo in the sentence coming after the diagram: horzontal should be horizontal. And maybe you want to write the last word of the proof as pro objects, since this is how you write it in the statement.

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