Lemma 15.91.1. 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.
$\square$

## Comments (1)

Comment #230 by Pieter Belmans on