Proposition 19.2.5. Let $R$ be a ring. Let $M$ be an $R$-module. Let $\kappa$ the cardinality of the set of submodules of $M$. If $\alpha$ is an ordinal whose cofinality is bigger than $\kappa$, then $M$ is $\alpha$-small with respect to injections.

Proof. The proof is straightforward, but let us first think about a special case. If $M$ is finite, then the claim is that for any inductive system $\{ B_\beta \}$ with injections between them, parametrized by a limit ordinal, any map $M \to \mathop{\mathrm{colim}}\nolimits B_\beta$ factors through one of the $B_\beta$. And this we proved in Lemma 19.2.3.

Now we start the proof in the general case. 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, we know that

$\bigcup f^{-1}(B_\beta ) = f^{-1}\left(\bigcup B_\beta \right) = M.$

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 $B_{\widetilde{\alpha }}$. $\square$

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