The Stacks project

Proposition 19.11.5. Let $\mathcal{A}$ be a Grothendieck abelian category. Let $M$ be an object of $\mathcal{A}$. Let $\kappa = |M|$. If $\alpha $ is an ordinal whose cofinality is bigger than $\kappa $, then $M$ is $\alpha $-small with respect to injections.

Proof. Please compare with Proposition 19.2.5. We need only show that the map (19.2.0.1) is a surjection. Let $f : M \to \mathop{\mathrm{colim}}\nolimits B_\beta $ be a map. Consider the subobjects $\{ f^{-1}(B_\beta )\} $ of $M$, where $B_\beta $ is considered as a subobject of the colimit $B = \bigcup _\beta B_\beta $. If one of these, say $f^{-1}(B_\beta )$, fills $M$, then the map factors through $B_\beta $.

So suppose to the contrary that all of the $f^{-1}(B_\beta )$ were proper subobjects of $M$. However, because $\mathcal{A}$ has AB5 we have

\[ \mathop{\mathrm{colim}}\nolimits f^{-1}(B_\beta ) = f^{-1}\left(\mathop{\mathrm{colim}}\nolimits B_\beta \right) = M. \]

Now there are at most $\kappa $ different subobjects of $M$ that occur among the $f^{-1}(B_\alpha )$, by hypothesis. Thus we can find a subset $S \subset \alpha $ of cardinality at most $\kappa $ such that as $\beta '$ ranges over $S$, the $f^{-1}(B_{\beta '})$ range over all the $f^{-1}(B_\alpha )$.

However, $S$ has an upper bound $\widetilde{\alpha } < \alpha $ as $\alpha $ has cofinality bigger than $\kappa $. In particular, all the $f^{-1}(B_{\beta '})$, $\beta ' \in S$ are contained in $f^{-1}(B_{\widetilde{\alpha }})$. It follows that $f^{-1}(B_{\widetilde{\alpha }}) = M$. In particular, the map $f$ factors through $B_{\widetilde{\alpha }}$. $\square$


Comments (1)

Comment #9497 by on

I think the phrase “so suppose to the contrary that all of the were proper subobjects of ” may be deleted. The argument does not do a proof by contradiction.

For the slow-thinkers out there (like me myself) the justification of the sentence “we only need to show that the map (19.2.0.1) is a surjection” is because is left-exact (Homology, Lemma 12.5.8) and because of the following

Lemma. Suppose is an AB5 abelian category. If is a direct system in such that is injective for all , then is injective.

Proof. We may assume that is initial in (replace by the directed subset , which is cofinal, and apply Categories, Lemma 4.17.2). The components of the morphism of direct systems are all injective. Hence is injective.

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  • 8 comment(s) on Section 19.11: Injectives in Grothendieck categories

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