The Stacks project

Lemma 20.39.2. Let $\mathcal{A}$ be an abelian category. Let $f : M \to M$ be a morphism of $\mathcal{A}$. If $M[f^ n] = \mathop{\mathrm{Ker}}(f^ n : M \to M)$ stabilizes, then the inverse systems

\[ (M \xrightarrow {f^ n} M) \quad \text{and}\quad \mathop{\mathrm{Coker}}(f^ n : M \to M) \]

are pro-isomorphic in $D(\mathcal{A})$.

Proof. There is clearly a map from the first inverse system to the second. Suppose that $M[f^ c] = M[f^{c + 1}] = M[f^{c + 2}] = \ldots $. Then we can define an arrow of inverse systems in $D(\mathcal{A})$ in the other direction by the diagrams

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

Since the top horizontal arrow is injective the complex in the top row is quasi-isomorphic to $\mathop{\mathrm{Coker}}(f^{n + c} : M \to M)$. Some details omitted. $\square$

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