## 19.10 Grothendieck's AB conditions

This and the next few sections are mostly interesting for “big” abelian categories, i.e., those categories listed in Categories, Remark 4.2.2. A good case to keep in mind is the category of sheaves of modules on a ringed site.

Grothendieck proved the existence of injectives in great generality in the paper [Tohoku]. He used the following conditions to single out abelian categories with special properties.

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

$\mathcal{A}$ has direct sums,

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

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

Here are the dual notions

$\mathcal{A}$ has products,

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

$\mathcal{A}$ 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.

Discussion: A direct sum in an abelian category is a coproduct. If an abelian category has direct sums (i.e., AB3), then it has colimits, see Categories, Lemma 4.14.12. Similarly if $\mathcal{A}$ has AB3* then it has limits, see Categories, Lemma 4.14.11. Exactness of direct sums means the following: given an index set $I$ and short exact sequences

\[ 0 \to A_ i \to B_ i \to C_ i \to 0,\quad i \in I \]

in $\mathcal{A}$ then the sequence

\[ 0 \to \bigoplus \nolimits _{i \in I} A_ i \to \bigoplus \nolimits _{i \in I} B_ i \to \bigoplus \nolimits _{i \in I} C_ i \to 0 \]

is exact as well. Without assuming AB4 it is only true in general that the sequence is exact on the right (i.e., taking direct sums is a right exact functor if direct sums exist). Similarly, exactness of filtered colimits means the following: given a directed set $I$ and a system of short exact sequences

\[ 0 \to A_ i \to B_ i \to C_ i \to 0 \]

over $I$ in $\mathcal{A}$ then the sequence

\[ 0 \to \mathop{\mathrm{colim}}\nolimits _{i \in I} A_ i \to \mathop{\mathrm{colim}}\nolimits _{i \in I} B_ i \to \mathop{\mathrm{colim}}\nolimits _{i \in I} C_ i \to 0 \]

is exact as well. Without assuming AB5 it is only true in general that the sequence is exact on the right (i.e., taking colimits is a right exact functor if colimits exist). A similar explanation holds for AB4* and AB5*.

## Comments (2)

Comment #2315 by Linyuan Liu on

Comment #2391 by Johan on