Lemma 10.82.14. Let R be a ring and let M \to M' be a map of R-modules. If M' is flat, then M \to M' is universally injective if and only if M/IM \to M'/IM' is injective for every finitely generated ideal I of R.
Proof. It suffices to show that M \otimes _ R Q \to M' \otimes _ R Q is injective for every finite R-module Q, see Theorem 10.82.3. Then Q has a finite filtration 0 = Q_0 \subset Q_1 \subset \ldots \subset Q_ n = Q by submodules whose subquotients are isomorphic to cyclic modules R/I_ i, see Lemma 10.5.4. Since M' is flat, we obtain a filtration
\xymatrix{ M \otimes Q_1 \ar[r] \ar[d] & M \otimes Q_2 \ar[r] \ar[d] & \ldots \ar[r] & M \otimes Q \ar[d] \\ M' \otimes Q_1 \ar@{^{(}->}[r] & M' \otimes Q_2 \ar@{^{(}->}[r] & \ldots \ar@{^{(}->}[r] & M' \otimes Q }
of M' \otimes _ R Q by submodules M' \otimes _ R Q_ i whose successive quotients are M' \otimes _ R R/I_ i = M'/I_ iM'. A simple induction argument shows that it suffices to check M/I_ i M \to M'/I_ i M' is injective. Note that the collection of finitely generated ideals I'_ i \subset I_ i is a directed set. Thus M/I_ iM = \mathop{\mathrm{colim}}\nolimits M/I'_ iM is a filtered colimit, similarly for M', the maps M/I'_ iM \to M'/I'_ i M' are injective by assumption, and since filtered colimits are exact (Lemma 10.8.8) we conclude. \square
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