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> [!theorem] Theorem. ([[Cohen-Seidenberg Theorems]])
>
Here we collect the **Cohen-Seidenberg Theorems** so that they may be referenced all at once as needed.[^1]
>- [[incomparability]]
>- [[lying over]]
>- [[going up]]
>- [[going down]]
>
A remark on notation: for [[ring|rings]] $A \subset B$ [[prime ideal|and]] $\mathfrak{p} \in \text{Spec }A$, we have as [[localization|as usual]] $A_{\mathfrak{p}}:=(A-\mathfrak{p}) ^{-1}A$. These notes also use the notation $B_{\mathfrak{p}}:=(A-\mathfrak{p})^{-1}A$, but note that $B_{\mathfrak{p}}$ is not a "localization at a prime ideal of $B
quot;. Importantly, $B_{\mathfrak{p}}$ is not usually a [[local ring]].
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####
[^1]: The first three of these take roughly the same amount of work to prove. The last one, [[going down]], is trickier.
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#### 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
> ```