The Stacks project

Lemma 10.88.5. Let $f: M \to N$ and $g: M \to M'$ be maps of $R$-modules. Suppose $\mathop{\mathrm{Coker}}(f)$ is of finite presentation. Then $g$ dominates $f$ if and only if $g$ factors through $f$, i.e. there exists a module map $h: N \to M'$ such that $g = h \circ f$.

Proof. Consider the pushout of $f$ and $g$ as in the statement of Lemma 10.88.4. From the construction of the pushout it follows that $\mathop{\mathrm{Coker}}(f') = \mathop{\mathrm{Coker}}(f)$, so $\mathop{\mathrm{Coker}}(f')$ is of finite presentation. Then by Lemma 10.82.4, $f'$ is universally injective if and only if

\[ 0 \to M' \xrightarrow {f'} N' \to \mathop{\mathrm{Coker}}(f') \to 0 \]

splits. This is the case if and only if there is a map $h' : N' \to M'$ such that $h' \circ f' = \text{id}_{M'}$. From the universal property of the pushout, the existence of such an $h'$ is equivalent to $g$ factoring through $f$. $\square$

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