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

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