Section 19.10 (079A): Grothendieck's AB conditions

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 (5)

Comment #2315 by Linyuan Liu on December 07, 2016 at 16:16

I'm currently reading the Tohoku paper. In the original paper the AB5 condition states that filtered union of sub objects distributes over intersection with another sub object. I've been thinking for several days of why this condition implies exactness of filtered colimits. Is there any references concerning this? Perhaps it's a stupid question...

Comment #2391 by Johan on February 17, 2017 at 00:44

This is not a stupid question. First of all, he says AB3 should be satisfied and the thing you say. In particular Grothendieck's AB5 implies the existence of colimits. Now let's for example try to show that colimits over $\mathbf{N}$ are exact if Grothendieck's AB5 holds. So assume given a short exact sequence of systems $0 \to (A_n) \to (B_n) \to (C_n) \to 0$ over $\mathbf{N}$ . It is clear that $\colim A_n \to \colim B_n \to \colim C_n \to 0$ is exact by looking at Homs into another object and using the mapping property of $\colim$ . OK, now let $K \subset \colim A_n$ be the kernel of the first map. Let $A'_n \subset \colim A_n$ be the image of $A_n$ . Observe that $\colim A_n = \sum A'_n$ as the map $\bigoplus A_n \to \colim A_n$ is surjective. Then Grothendieck's axiom AB5 says it suffices to show that $K \cap A'_n = 0$ . I claim that $\Ker(A_n \to A'_n) = \bigcup \Ker(A_n \to A_{n + m})$ . To see this I suggest you think about the exact sequence $0 \to \bigoplus A_n \to \bigoplus A_n \to \colim A_n \to 0$ and use Grothendieck's axiom AB5 about unions inside the object in the middle. A similar argument shows that the inverse image of $K$ in $A_n$ is $\bigcup \Ker(A_n \to B_{n + m})$ . Since the maps $A_{n + m} \to B_{n + m}$ are injective we conclude. Presumably there are shorter proofs as well.

Anyway, it doesn't matter as we will use AB5 as stated throughout the document.

Comment #9372 by Elías Guisado on June 05, 2024 at 02:44

For those whose browser does not compile well the LaTeX in comment #2391 (like mine), you may find #2391 in plain text here.

@#2315 A proof of the equivalence between definitions of the AB5 condition is N. Popescu, Abelian Categories with Applications to Rings and Modules, Sect. 2.8, Theorem 8.6.

In case it is interesting to point this out: AB5 implies AB4 as it is currently stated in 19.10.1. A reference is ibid., Theorem 8.8.

Comment #9401 by Elías Guisado on June 05, 2024 at 10:29

I just realized about this quick proof of AB5 $\Rightarrow$ AB4:

If our category is AB3 (i.e., all coproducts exist; thus it is cocomplete), then we can write [ref] $\label{1}\tag{1} \coprod_{i\in I}X_i=\underset{F\in T}{\operatorname{colim}}\coprod_{j\in F}X_j,$ where $T$ is the set of finite subsets of $I$ , which is a directed set. On the one hand, since finite coproducts are also products in a preadditive category (Homology, Lemma 12.3.4), they are exact (i.e., they commute with all finite limits and colimits). Thus, if our category is AB5, then the expression \eqref{1} is exact in $\{X_i\}_{i\in I}$ , i.e., $A_i\xrightarrow{f_i} B_i\xrightarrow{g_i} C_i$ exact for all $i$ implies $\coprod A_i\to\coprod B_i\to\coprod C_i$ exact.

Comment #9427 by Stacks project on June 06, 2024 at 08:05

Going to leave as is.

The Stacks project

19.10 Grothendieck's AB conditions

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