----- > [!proposition] Proposition. ([[flatness is a local property]]) > Let $R$ be a (say, [[commutative ring|commutative]]) [[ring]] and $M$ an $R$-[[module]]. The following are equivalent: > >1. $M$ is [[flat module|flat]] >2. $M_{\mathfrak{p}}$ is [[flat module|flat]] for all $\mathfrak{p} \in \text{Spec }R$ >3. $M_{\mathfrak{m}}$ is [[flat module|flat]] for all $\mathfrak{m} \in \text{mSpec }R$. ^proposition > [!proof]- Proof. ([[flatness is a local property]]) > ~ ----- #### ---- #### 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 > ```