Lemma 12.31.5. Let $\mathcal{A}$ be an abelian category. Let $(A_ i)$ be an inverse system in $\mathcal{A}$ with limit $A = \mathop{\mathrm{lim}}\nolimits A_ i$. Then $(A_ i)$ is essentially constant (see Categories, Definition 4.22.1) if and only if there exists an $i$ and for all $j \geq i$ a direct sum decomposition $A_ j = A \oplus Z_ j$ such that (a) the maps $A_{j'} \to A_ j$ are compatible with the direct sum decompositions, (b) for all $j$ there exists some $j' \geq j$ such that $Z_{j'} \to Z_ j$ is zero.

Proof. Assume $(A_ i)$ is essentially constant. Then there exists an $i$ and a morphism $A_ i \to A$ such that $A \to A_ i \to A$ is the identity and for all $j \geq i$ there exists a $j' \geq j$ such that $A_{j'} \to A_ j$ factors as $A_{j'} \to A_ i \to A \to A_ j$ (the last map comes from $A = \mathop{\mathrm{lim}}\nolimits A_ i$). Hence setting $Z_ j = \mathop{\mathrm{Ker}}(A_ j \to A)$ for all $j \geq i$ works. Proof of the converse is omitted. $\square$

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