----- > [!proposition] Proposition. ([[being zero is a local property]]) > For an $R$-[[module]] $M$, the following are equivalent: > 1. $M=0$ > 2. $M_{\mathfrak{p}}=0$ for each [[prime ideal]] $\mathfrak{p}$ of $R$ > 3. $M_\mathfrak{m}=0$ for each [[maximal ideal]] $\mathfrak{m}$ of $R$ > > [!proposition] Corollary. > Injectivity and surjectivity are local properties (apply the proposition to $\operatorname{ker }$ and $\operatorname{coker }$). ^proposition > [!proof]- Proof. ([[being zero is a local property]]) > $1 \implies 2 \implies 3$ is clear. Assume (3). Take $m \in M$. Consider the [[annihilator of a module|annihilator ideal]] $\text{Ann}_{R}(m)=\{ r \in R: rm=0 \}$. It suffices to show $\text{Ann}_{R}(M)=R$, because then $1 \in \text{Ann}_{R}(M)$ and so $M=0$. So it suffices to prove that $\text{Ann}_{R}(M)$ is not contained in any $\mathfrak{m} \in \text{mSpec }R$. Fix $\mathfrak{m} \in \text{mSpec }R$. Since $M_{\mathfrak{m}}=0$, $\frac{m}{1}=\frac{0}{1}$ and so $um=0$ for some $u \not \in \mathfrak{m}$, i.e. $u \in \text{Ann}_{R}(m)-\mathfrak{m}$ and so $\text{Ann}_{R}(m) \not \subset \mathfrak{m}$. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```