The Stacks project

Definition 19.10.1. Let $\mathcal{A}$ be an abelian category. We name some conditions

  1. $\mathcal{A}$ has direct sums,

  2. $\mathcal{A}$ has AB3 and direct sums are exact,

  3. $\mathcal{A}$ has AB3 and filtered colimits are exact.

Here are the dual notions

  1. $\mathcal{A}$ has products,

  2. $\mathcal{A}$ has AB3* and products are exact,

  3. $\mathcal{A}$ has AB3* and cofiltered limits are exact.

We say an object $U$ of $\mathcal{A}$ is a generator if for every $N \subset M$, $N \not= M$ in $\mathcal{A}$ there exists a morphism $U \to M$ which does not factor through $N$. We say $\mathcal{A}$ is a Grothendieck abelian category if it has AB5 and a generator.

Comments (2)

Comment #7527 by Samuel Tiersma on

It seems to me AB5* should state 'cofiltered limits' are exact (since by Lemma 4.12.5 limits of directed inverse systems coincide with cofiltered limits).

There are also:

  • 2 comment(s) on Section 19.10: Grothendieck's AB conditions

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Your email address will not be published.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 079B. Beware of the difference between the letter 'O' and the digit '0'.