The Stacks project

Lemma 47.2.2. Let $\mathcal{A}$ be an abelian category.

  1. If $A \subset B$ and $B \subset C$ are essential extensions, then $A \subset C$ is an essential extension.

  2. If $A \subset B$ is an essential extension and $C \subset B$ is a subobject, then $A \cap C \subset C$ is an essential extension.

  3. If $A \to B$ and $B \to C$ are essential surjections, then $A \to C$ is an essential surjection.

  4. Given an essential surjection $f : A \to B$ and a surjection $A \to C$ with kernel $K$, the morphism $C \to B/f(K)$ is an essential surjection.

Proof. Omitted. $\square$

Comments (0)

There are also:

  • 4 comment(s) on Section 47.2: Essential surjections and injections

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Your email address will not be published.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 08XK. Beware of the difference between the letter 'O' and the digit '0'.