Lemma 38.7.5. Let (R, \mathfrak m) be a local ring. Let u : M \to N be an R-module map. If M is a projective R-module, N is a flat R-module, and \overline{u} : M/\mathfrak mM \to N/\mathfrak mN is injective then u is universally injective.
Proof. By Algebra, Theorem 10.85.4 the module M is free. If we show the result holds for every finitely generated direct summand of M, then the lemma follows. Hence we may assume that M is finite free. Write N = \mathop{\mathrm{colim}}\nolimits _ i N_ i as a directed colimit of finite free modules, see Algebra, Theorem 10.81.4. Note that u : M \to N factors through N_ i for some i (as M is finite free). Denote u_ i : M \to N_ i the corresponding R-module map. As \overline{u} is injective we see that \overline{u_ i} : M/\mathfrak mM \to N_ i/\mathfrak mN_ i is injective and remains injective on composing with the maps N_ i/\mathfrak mN_ i \to N_{i'}/\mathfrak mN_{i'} for all i' \geq i. As M and N_{i'} are finite free over the local ring R this implies that M \to N_{i'} is a split injection for all i' \geq i. Hence for any R-module Q we see that M \otimes _ R Q \to N_{i'} \otimes _ R Q is injective for all i' \geq i. As - \otimes _ R Q commutes with colimits we conclude that M \otimes _ R Q \to N_{i'} \otimes _ R Q is injective as desired. \square
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