Lemma 38.7.3. Let $R \to S$ be a ring map. Let $N$ be an $S$-module. Let $S \to S'$ be a ring map. Assume

1. $R \to S$ is a local homomorphism of local rings

2. $S$ is essentially of finite presentation over $R$,

3. $N$ is of finite presentation over $S$,

4. $N$ is flat over $R$,

5. $S \to S'$ is flat, and

6. the image of $\mathop{\mathrm{Spec}}(S') \to \mathop{\mathrm{Spec}}(S)$ contains all primes $\mathfrak q$ of $S$ lying over $\mathfrak m_ R$ such that $\mathfrak q$ is an associated prime of $N/\mathfrak m_ R N$.

Then $N \to N \otimes _ S S'$ is $R$-universally injective.

Proof. Set $N' = N \otimes _ R S'$. Consider the commutative diagram

$\xymatrix{ N \ar[d] \ar[r] & N' \ar[d] \\ \Sigma ^{-1}N \ar[r] & \Sigma ^{-1}N' }$

where $\Sigma \subset S$ is the set of elements which are not a zerodivisor on $N/\mathfrak m_ R N$. If we can show that the map $N \to \Sigma ^{-1}N'$ is universally injective, then $N \to N'$ is too (see Algebra, Lemma 10.82.10).

By Lemma 38.7.1 the ring $\Sigma ^{-1}S$ is a semi-local ring whose maximal ideals correspond to associated primes of $N/\mathfrak m_ R N$. Hence the image of $\mathop{\mathrm{Spec}}(\Sigma ^{-1}S') \to \mathop{\mathrm{Spec}}(\Sigma ^{-1}S)$ contains all these maximal ideals by assumption. By Algebra, Lemma 10.39.16 the ring map $\Sigma ^{-1}S \to \Sigma ^{-1}S'$ is faithfully flat. Hence $\Sigma ^{-1}N \to \Sigma ^{-1}N'$, which is the map

$N \otimes _ S \Sigma ^{-1}S \longrightarrow N \otimes _ S \Sigma ^{-1}S'$

is universally injective, see Algebra, Lemmas 10.82.11 and 10.82.8. Finally, we apply Lemma 38.7.2 to see that $N \to \Sigma ^{-1}N$ is universally injective. As the composition of universally injective module maps is universally injective (see Algebra, Lemma 10.82.9) we conclude that $N \to \Sigma ^{-1}N'$ is universally injective and we win. $\square$

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