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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