The Stacks project

108.57 A big abelian category

The purpose of this section is to give an example of a “big” abelian category $\mathcal{A}$ and objects $M, N$ such that the collection of isomorphism classes of extensions $\mathop{\mathrm{Ext}}\nolimits _\mathcal {A}(M, N)$ is not a set. The example is due to Freyd, see [page 131, Exercise A, Freyd].

We define $\mathcal{A}$ as follows. An object of $\mathcal{A}$ consists of a triple $(M, \alpha , f)$ where $M$ is an abelian group and $\alpha $ is an ordinal and $f : \alpha \to \text{End}(M)$ is a map. A morphism $(M, \alpha , f) \to (M', \alpha ', f')$ is given by a homomorphism of abelian groups $\varphi : M \to M'$ such that for any ordinal $\beta $ we have

\[ \varphi \circ f(\beta ) = f'(\beta ) \circ \varphi \]

Here the rule is that we set $f(\beta ) = 0$ if $\beta $ is not in $\alpha $ and similarly we set $f'(\beta )$ equal to zero if $\beta $ is not an element of $\alpha '$. We omit the verification that the category so defined is abelian.

Consider the object $Z = (\mathbf{Z}, \emptyset , f)$, i.e., all the operators are zero. The observation is that computed in $\mathcal{A}$ the group $\mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {A}(Z, Z)$ is a proper class and not a set. Namely, for each ordinal $\alpha $ we can find an extension $(M, \alpha + 1, f)$ of $Z$ by $Z$ whose underlying group is $M = \mathbf{Z} \oplus \mathbf{Z}$ and where the value of $f$ is always zero except for

\[ f(\alpha ) = \left( \begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix} \right). \]

This clearly produces a proper class of isomorphism classes of extensions. In particular, the derived category of $\mathcal{A}$ has proper classes for its collections of morphism, see Derived Categories, Lemma 13.27.6. This means that some care has to be exercised when defining Verdier quotients of triangulated categories.

Lemma 108.57.1. There exists a “big” abelian category $\mathcal{A}$ whose $\mathop{\mathrm{Ext}}\nolimits $-groups are proper classes.

Proof. See discussion above. $\square$


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