Lemma 10.89.4. Let M be an R-module, P a finitely presented R-module, and f: P \to M a map. Let Q be an R-module and suppose x \in \mathop{\mathrm{Ker}}(P \otimes Q \to M \otimes Q). Then there exists a finitely presented R-module P' and a map f': P \to P' such that f factors through f' and x \in \mathop{\mathrm{Ker}}(P \otimes Q \to P' \otimes Q).
Proof. Write M as a colimit M = \mathop{\mathrm{colim}}\nolimits _{i \in I} M_ i of a directed system of finitely presented modules M_ i. Since P is finitely presented, the map f: P \to M factors through M_ j \to M for some j \in I. Upon tensoring by Q we have a commutative diagram
\xymatrix{ & M_ j \otimes Q \ar[dr] & \\ P \otimes Q \ar[ur] \ar[rr] & & M \otimes Q . }
The image y of x in M_ j \otimes Q is in the kernel of M_ j \otimes Q \to M \otimes Q. Since M \otimes Q = \mathop{\mathrm{colim}}\nolimits _{i \in I} (M_ i \otimes Q), this means y maps to 0 in M_{j'} \otimes Q for some j' \geq j. Thus we may take P' = M_{j'} and f' to be the composite P \to M_ j \to M_{j'}. \square
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