The Stacks project

Lemma 12.3.16. Let $\mathcal{A}$ be a preadditive category, $x$ be an object of $\mathcal{A}$, and $f : x \to x$ be idempotent. Write $g = \text{id} _ x - f$. If each of $f$ and $g$ have at least one of a kernel or cokernel and in turn morphisms $i , j , p , q$ are constructed as in the statement of Lemma 12.3.15, then all four (co)kernels exist and

\begin{align*} x & \simeq \mathop{\mathrm{Ker}}(f) \oplus \mathop{\mathrm{Ker}}(g) \\ & \simeq \mathop{\mathrm{Coker}}(f) \oplus \mathop{\mathrm{Ker}}(g) \\ & \simeq \mathop{\mathrm{Ker}}(f) \oplus \mathop{\mathrm{Coker}}(g) \\ & \simeq \mathop{\mathrm{Coker}}(f) \oplus \mathop{\mathrm{Coker}}(g) \end{align*}

with the structure of each direct product given by $i , j , p , q$.

Proof. By Lemma 12.3.15 $p \circ i = \text{id} _{\mathop{\mathrm{Ker}}(f)} = \text{id} _{\mathop{\mathrm{Coker}}(f)}$ and $q \circ j = \text{id} _{\mathop{\mathrm{Ker}}(g)} = \text{id} _{\mathop{\mathrm{Coker}}(g)}$. By construction, $i \circ p + j \circ q = f + g = \text{id} _ x$. By Remark 12.3.6 the claim follows. $\square$

Comments (0)

There are also:

  • 10 comment(s) on Section 12.3: Preadditive and additive categories

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Your email address will not be published.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 0H9Q. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 12.3.16 as pdf