---- > [!definition] Definition. ([[Krull dimension]]) > The **(Krull) dimension** of a [[commutative ring|commutative]] [[ring]] $A$, $\text{dim }A$, is $\sup \{ \text{ht }\mathfrak{p} : \mathfrak{p} \text{ is a prime ideal of }A \},$ where $\text{ht }\mathfrak{p}$ denotes the [[height of a prime ideal]]. By convention, the dimension of the zero ring is defined to be $-1$ or $-\infty$. ^definition > [!equivalence] > - Easy to see: $\text{dim }A=\sup\{ \text{ht }\mathfrak{m}: \mathfrak{m} \in \text{mSpec }A \}$. > - Equivalently, $\dim A=\dim \text{Spec } A$, the [[dimension of a topological space|dimension]] of the [[prime ideal|spectrum of]] $A$ as a [[topological space]] (since [[irreducible closed subspaces of Spec are precisely the vanishing of primes]]). ^equivalence > [!basicproperties] > - $\text{dim }A_{\mathfrak{p}}=\text{ht }\mathfrak{p}$ ([[height of a prime ideal#^71f2b0|see]]). > - $\text{dim }A=\sup\{ \text{dim }A_{\mathfrak{m}}: \mathfrak{m} \in \text{mSpec }A \}$ (follows from above). Thus, the computation of the dimension of a general ring reduces to the computations of the dimensions of [[local ring|local rings]]. > - If $A \subset B$ is an [[integral algebra|integral extension]] of [[ring|rings]], then $\text{dim }A=\text{dim }B$. See [[integral extensions are zero-dimensional]]. If $A,B$ are also $k$-[[algebra|algebras]], $k$ a [[field]], then $\text{trdeg}_{k}A=\text{trdeg}_{k}B$. > - The dimension of a [[subalgebra generated by a subset|finite type]] $k$-[[algebra]] is the cardinality of the $k$-[[algebraically independent]] subset provided by [[Noether's normalization theorem]] (intuitive) [^1] > - If $A$ is a $k$-[[algebra]] [[subalgebra generated by a subset|of finite type]], and also an [[integral domain]], [[transcendence basis|then]][^2] $\text{dim }A=\text{trdeg}_{k}A.$ ^properties > [!basicexample] > > - If $k$ is a [[field]], then $\dim k[T_{1},\dots,T_{n}] \geq n$ (this is, in fact, in equality — will see later in the course). Indeed, the chain of ideals $\begin{align} > \langle T_{1}-a_{1},\dots,T_{n}-a_{n} \rangle &\supsetneq \langle T_{1} - a_{1},\dots ,T_{n-1}-a_{n-1} \rangle \\ > & \supsetneq \cdots \supsetneq \langle T_{1}-a_{1} \rangle \supsetneq (0) > \end{align}$ > witnesses as such. > >- Any [[PID]] is one-dimensional. For instance, $\mathbb{Z}$ is one-dimensional (as one would hope). So is $k[T]$. ---- #### [^1]: We know this once we've shown $\text{dim }k[T_{1},\dots,T_{n}]=n$. NNT gives a [[polynomial 4|polynomial]] subalgebra $k[x_{1},\dots,x_{n}]$ over which $A$ is [[finite algebra|finite]], [[integral algebra|hence integral]], and [[integral extensions are zero-dimensional]]. [^2]: This also is from [[Noether's normalization theorem|NNT]], as well as [[integral extensions are zero-dimensional|integral extension of k-algebras preserves transcendence degree]]. $k[x_{1},\dots,x_{n}] \hookrightarrow A$ preserves $\text{trdeg}$... so $\text{dim }A=\underbrace{ \text{dim }k[T_{1},\dots,T_{n}] }_{ =d }$and $\text{trdeg}_{k}A=\underbrace{ \text{trdeg}_{k}k[T_{1},\dots,T_{d}] }_{ =d }.$ ---- #### 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 > ```