----- > [!proposition] Proposition. ([[integral extensions are zero-dimensional]]) > Let $A \subset B$ be an [[integral algebra|integral extension]]. Then their [[Krull dimension|dimensions]] agree: $\text{dim }A=\text{dim }B.$ If the extension is in fact of $k$-[[algebra|algebras]], with $A,B$ [[integral domain|integral domains]] and $k$ some [[field]], then the [[transcendence basis|transcendence degrees]] agree: $\text{trdeg}_{k}A=\text{trdeg}_{k}B$ ^proposition > [!proof]- Proof. ([[integral extensions are zero-dimensional]]) > > Summary: > 1. Draw a picture with $\text{Spec }B$ hanging over $\text{Spec }A$. > 2. Using [[lying over]] and [[going up]] to show $\text{dim }A \leq \text{dim }B$ > 3. To show $\text{dim }B \geq \text{dim }A$, just push a chain in $B$ through $\iota^{*}$. [[Incomparability]] guarantees that it will yield a chain in $A$. > > **$\text{dim }A \leq \text{dim }B.$** > > Take a chain $\mathfrak{p}_{0} \subsetneq \dots \subsetneq \mathfrak{p}_{d}, \mathfrak{p}_{i} \in \text{Spec }A, d \geq 0.$ > By [[lying over]] and [[going up]], there are $\mathfrak{q}_{1} \subset \dots \subset \mathfrak{q}_{d}$, $\mathfrak{q}_{i} \in \text{Spec }B$, such that $\mathfrak{q}_{i} \cap A=\mathfrak{p}_{i}$. We must have $\mathfrak{q}_{i} \neq \mathfrak{q}_{i+1}$ because $\mathfrak{p}_{i} \neq \mathfrak{p}_{i+1}$. Thus $\text{dim }B \geq d$, thus $\text{dim }B \geq \text{dim }A$ since $d$ was arbitrary. > > **$\text{dim }B \leq \text{dim }A$.** Take a chain $\mathfrak{q}_{0} \subsetneq \dots \subsetneq \mathfrak{q}_{d}, \mathfrak{q}_{i} \in \text{Spec }B, d \geq 0.$ > Contracting to $A$, get a chain $\mathfrak{q}_{0} \cap A \subset \dots \subset \mathfrak{q}_{d} \cap A.$ > We must have $\mathfrak{q}_{i} \cap A \neq \mathfrak{q}_{i+1} \cap A$ because $\mathfrak{q }_{i}\subsetneq \mathfrak{q}_{i+1}$ and [[incomparability]]. Thus, $\text{dim }A \geq d$, and thus $\text{dim }A \geq \text{dim }B$ since $d$ was arbitrary. ----- #### ---- #### 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 > ```