Lemma 10.22.1. Let $R$ be a ring.

For an element $x$ of an $R$-module $M$ the following are equivalent

$x = 0$,

$x$ maps to zero in $M_\mathfrak p$ for all $\mathfrak p \in \mathop{\mathrm{Spec}}(R)$,

$x$ maps to zero in $M_{\mathfrak m}$ for all maximal ideals $\mathfrak m$ of $R$.

In other words, the map $M \to \prod _{\mathfrak m} M_{\mathfrak m}$ is injective.

Given an $R$-module $M$ the following are equivalent

$M$ is zero,

$M_{\mathfrak p}$ is zero for all $\mathfrak p \in \mathop{\mathrm{Spec}}(R)$,

$M_{\mathfrak m}$ is zero for all maximal ideals $\mathfrak m$ of $R$.

Given a complex $M_1 \to M_2 \to M_3$ of $R$-modules the following are equivalent

$M_1 \to M_2 \to M_3$ is exact,

for every prime $\mathfrak p$ of $R$ the localization $M_{1, \mathfrak p} \to M_{2, \mathfrak p} \to M_{3, \mathfrak p}$ is exact,

for every maximal ideal $\mathfrak m$ of $R$ the localization $M_{1, \mathfrak m} \to M_{2, \mathfrak m} \to M_{3, \mathfrak m}$ is exact.

Given a map $f : M \to M'$ of $R$-modules the following are equivalent

$f$ is injective,

$f_{\mathfrak p} : M_\mathfrak p \to M'_\mathfrak p$ is injective for all primes $\mathfrak p$ of $R$,

$f_{\mathfrak m} : M_\mathfrak m \to M'_\mathfrak m$ is injective for all maximal ideals $\mathfrak m$ of $R$.

Given a map $f : M \to M'$ of $R$-modules the following are equivalent

$f$ is surjective,

$f_{\mathfrak p} : M_\mathfrak p \to M'_\mathfrak p$ is surjective for all primes $\mathfrak p$ of $R$,

$f_{\mathfrak m} : M_\mathfrak m \to M'_\mathfrak m$ is surjective for all maximal ideals $\mathfrak m$ of $R$.

Given a map $f : M \to M'$ of $R$-modules the following are equivalent

$f$ is bijective,

$f_{\mathfrak p} : M_\mathfrak p \to M'_\mathfrak p$ is bijective for all primes $\mathfrak p$ of $R$,

$f_{\mathfrak m} : M_\mathfrak m \to M'_\mathfrak m$ is bijective for all maximal ideals $\mathfrak m$ of $R$.

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