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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)