The Stacks project

Lemma 13.42.1. Let $\mathcal{D}$ be a triangulated category. Let $(A_ i)$ be an inverse system in $\mathcal{D}$. 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 and identity on $A$, (b) for all $j \geq i$ there exists some $j' \geq j$ such that $Z_{j'} \to Z_ j$ is zero.

Proof. Assume $(A_ i)$ is essentially constant with value $A$. Then $A = \mathop{\mathrm{lim}}\nolimits A_ i$ and there exists an $i$ and a morphism $A_ i \to A$ such that (1) the composition $A \to A_ i \to A$ is the identity on $A$ and (2) for all $j \geq i$ there exists a $j' \geq j$ such that $A_{j'} \to A_ j$ factors as $A_{j'} \to A_ i \to A \to A_ j$. From (1) we conclude that for $j \geq i$ the maps $A \to A_ j$ and $A_ j \to A_ i \to A$ compose to the identity on $A$. It follows that $A_ j \to A$ has a kernel $Z_ j$ and that the map $A \oplus Z_ j \to A_ j$ is an isomorphism, see Lemmas 13.4.12 and 13.4.11. These direct sum decompositions clearly satisfy (a). From (2) we conclude that for all $j$ there is a $j' \geq j$ such that $Z_{j'} \to Z_ j$ is zero, so (b) holds. Proof of the converse is omitted. $\square$

Comments (0)

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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0G39. Beware of the difference between the letter 'O' and the digit '0'.