Lemma 10.88.3. Let $f: M \to N$ and $g: M \to M'$ be maps of $R$-modules. Then $g$ dominates $f$ if and only if for any finitely presented $R$-module $Q$, we have $\mathop{\mathrm{Ker}}(f \otimes _ R \text{id}_ Q) \subset \mathop{\mathrm{Ker}}(g \otimes _ R \text{id}_ Q)$.

Proof. Suppose $\mathop{\mathrm{Ker}}(f \otimes _ R \text{id}_ Q) \subset \mathop{\mathrm{Ker}}(g \otimes _ R \text{id}_ Q)$ for all finitely presented modules $Q$. If $Q$ is an arbitrary module, write $Q = \mathop{\mathrm{colim}}\nolimits _{i \in I} Q_ i$ as a colimit of a directed system of finitely presented modules $Q_ i$. Then $\mathop{\mathrm{Ker}}(f \otimes _ R \text{id}_{Q_ i}) \subset \mathop{\mathrm{Ker}}(g \otimes _ R \text{id}_{Q_ i})$ for all $i$. Since taking directed colimits is exact and commutes with tensor product, it follows that $\mathop{\mathrm{Ker}}(f \otimes _ R \text{id}_ Q) \subset \mathop{\mathrm{Ker}}(g \otimes _ R \text{id}_ Q)$. $\square$

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