---- > [!definition] Definition. ([[variety]]) > Let $k$ be an [[algebraically closed]] [[field]]. A **variety over $k$** is an [[integral scheme|integral]] [[scheme]] $X$ ([[separated scheme morphism|separated]]?) [[scheme morphism locally of finite type|of finite type]] [[scheme over a field|over]] $k$. That is, a variety over $k$ is a [[scheme]] $X$ together with together with a [[morphism of locally ringed spaces|morphism]] $X \to \text{Spec }k$, such that $f$ is ([[separated scheme morphism|separated]]?) [[scheme morphism locally of finite type|of finite type]] and $X$ is [[integral scheme|integral]]. ^definition > [!basicnonexample] Note. > Removing 'separated' allows schemes like $\mathbb{A}^{1}$ with doubled origin to be varieties, which may or may not be desirable (usually not desirable). > [!NOTE] Remark. > It's easy to see the 'of finite type over $k condition is merely the following: there exists a finite [[cover]] of $X$ by [[affine scheme|open affines]] $U_{1},\dots,U_{n}$, $U_{i}=\text{Spec }A_{i}$, such that each $A_{i}$ is [[subalgebra generated by a subset|finitely generated]] as a $k$-[[algebra]]: $A_{i}=k[T_{1},\dots,T_{n}] / I$ for some [[ideal]] $I$. The $A_{i}$ must moreover be [[integral domain|integral domains]], so $I$ must moreover be [[prime ideal|prime]].[^1] > [!basicproperties] > For any [[closed set|closed point]] $p \in x$, $\text{dim }X=\text{dim }\mathcal{O}_{X, p}$. > > > [!proof]- > > > > > > **The affine case.** First assume $X=\text{Spec }A$ is an [[affine scheme|affine]] [[variety]], say, $p=\mathfrak{p}$ for some [[prime ideal]] $\mathfrak{p}$ of $A$. First note $\{ \mathfrak{p} \}$ [[closed set|closed]] [[Zariski topology on a ring spectrum|implies]] $\{ \mathfrak{p} \}= \overline{\{ \mathfrak{p} \}}= V(\mathfrak{p})$. Then: > > $\begin{align} > > \text{dim } \mathcal{O}_{X, \mathfrak{p}} &= \text{dim }A_{\mathfrak{p}} \\ > > &= \text{ht }\mathfrak{p} \\ > > &= \text{codim}(V(\mathfrak{p}), \text{Spec }A) \\ > > &= \text{codim}(\{ \mathfrak{p} \}, \text{Spec } A) \\ > > &= \text{dim }\text{Spec } A - \cancel{ \text{dim } \{ \mathfrak{p} \} }^{=0} \\ > > &= \text{dim } X > > \end{align}$ > > where: > > - Line 1 [[structure sheaf on a ring spectrum|uses that]] $\mathcal{O}_{X, \mathfrak{p}}=A_{\mathfrak{p}}$; > > - Line 2 uses [[height of a prime ideal|the basic property here]]; > > - Line 3 uses the [[codimension of a closed subspace|example here]]; > > - Line 4 uses that $\{ \mathfrak{p} \}=V(\mathfrak{p})$ since $\mathfrak{p}$ is a closed point; > > - Line 5 uses that [[codimension is well-behaved for irreducible closed subspaces of varieties over a field]]. > > > > > > **The general case.** $X$ is [[integral scheme|integral]], hence [[irreducible element of an integral domain|irreducible]], and so for any open affine $U \subset X$ we have $\text{dim }U=\text{dim }X$ per the properties in [[irreducible topological space]]. This reduces us to the affine case. > > > > > > > [^1]: And so each $A_{i}$ corresponds to an [[affine variety]], per [[Hilbert's geometry-algebra correspondence]]. ---- #### ---- #### 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 > ```