The Stacks project

Lemma 19.12.1. Let $\mathcal{A}$ be a Grothendieck abelian category with generator $U$. Let $c$ be the function on cardinals defined by $c(\kappa ) = |\bigoplus _{\alpha \in \kappa } U|$. If $\pi : M \to N$ is a surjection then there exists a subobject $M' \subset M$ which surjects onto $N$ with $|M'| \leq c(|N|)$.

Proof. For every proper subobject $N' \subset N$ choose a morphism $\varphi _{N'} : U \to M$ such that $U \to M \to N$ does not factor through $N'$. Set

\[ M' = \mathop{\mathrm{Im}}\left( \bigoplus \nolimits _{N' \subset N} \varphi _{N'} : \bigoplus \nolimits _{N' \subset N} U \longrightarrow M\right) \]

Then $M'$ works. $\square$

Comments (2)

Comment #8034 by François Loeser on

There is a typo on line 3: should be

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