Dimension Constructible Images and Dimensions of Fibres

Given: The objects and hypotheses in the statement.

For a Noetherian topological space $T$, define $\dim T$ as the supremum of the lengths $s$ of strict chains $Z_0⊊\cdots ⊊Z_s$ of nonempty irreducible closed subsets of $T$. Thus a one-member chain has length zero. Set $\dim \mathrm{\varnothing }=-\mathrm{\infty }$, and allow $\dim T=+\mathrm{\infty }$. The supremum of an empty family of dimensions is $-\mathrm{\infty }$. (Chain dimension and the empty-space convention).

Given: The objects and hypotheses in the statement.

For a Noetherian topological space $T$, define $\dim T$ as the supremum of the lengths $s$ of strict chains $Z_0⊊\cdots ⊊Z_s$ of nonempty irreducible closed subsets of $T$. Thus a one-member chain has length zero. Set $\dim \mathrm{\varnothing }=-\mathrm{\infty }$, and allow $\dim T=+\mathrm{\infty }$. The supremum of an empty family of dimensions is $-\mathrm{\infty }$. (Chain dimension and the empty-space convention).

Given: The objects and hypotheses in the statement.

Fix an algebraically closed field $k$. A classical algebraic prevariety over $k$ is a quasi-compact locally ringed space over $k$ that is covered by open subspaces isomorphic, as locally ringed spaces over $k$, to affine algebraic sets with their regular-function sheaves. Equivalently, it admits a finite such affine cover. A map is regular when it is a morphism of these locally ringed spaces over $k$; equivalently, this can be checked on affine charts. Thus its maps on structure sheaves respect the fixed $k$-algebra structures. A prevariety is separated when the equalizer of every pair of regular maps into it is closed. A (classical) algebraic variety is a separated prevariety. In the comparison below, “irreducible classical variety” means a nonempty variety whose underlying topological space is irreducible. (Classical algebraic prevarieties, regular maps, and varieties).

Let $(X,\mathcal{T})$ be a topological space (def-topological-space). The space $X$ is Noetherian when every ascending chain $U_0\subseteq U_1\subseteq U_2\subseteq \cdots$ of open subsets stabilizes. Equivalently, $X$ is Noetherian when every descending chain $F_0\supseteq F_1\supseteq F_2\supseteq \cdots$ of closed subsets stabilizes. (Noetherian topological spaces via ACC on opens or DCC on closed subsets).

Let $k$ be an algebraically closed field and let $X\subseteq {\mathbf{A}}_k^n$ be an affine algebraic set. Then there exist finitely many irreducible closed subsets $X_1,\dots ,X_m\subseteq X$ such that $X=X_1\cup \cdots \cup X_m,$ no $X_i$ is contained in the union of the others, and the family $\{X_1,\dots ,X_m\}$ is uniquely determined up to reordering. These sets are the irreducible components of $X$. (Every affine algebraic set has finitely many irreducible components).

Let $R$ be a Noetherian commutative ring. Then the polynomial ring $R[x]$ is a Noetherian commutative ring. No hypothesis beyond Noetherianity is placed on $R$: it may have zero divisors, and it may be the zero ring. (Hilbert basis theorem: if $R$ is Noetherian then $R[x]$ is Noetherian).

Given: The objects and hypotheses in the statement.

For a classical variety $X$, let $\dim X$ be its chain dimension. If $X_1,\dots ,X_m$ are its irreducible components and $x\in X$ is a closed point, define ${\dim }_{x}X={\max }_{x\in X_i}\dim X_i$. The indexing family is nonempty. Say that $X$ has pure dimension $d$ if every irreducible component has dimension $d$; the condition on components is vacuous for the empty variety, whose dimension is nevertheless $-\mathrm{\infty }$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Global and local dimension of classical varieties).

Assume the Axiom of Choice. Let $k$ be an algebraically closed field. 1. The assignments $X⟼I(X),J⟼V(J)$ induce mutually inverse inclusion-reversing correspondences between affine algebraic sets $X\subseteq {\mathbf{A}}_k^n$ and radical ideals $J\subseteq k[x_1,\dots ,x_n]$. 2. Under this correspondence, nonempty irreducible affine algebraic sets correspond exactly to prime ideals. (Affine algebraic sets correspond to radical ideals, and irreducible ones to prime ideals).

Let $R$ be a nonzero commutative ring. A strict chain of prime ideals of length $n$ is a sequence ${\mathfrak{p}}_0⊊{\mathfrak{p}}_1⊊\cdots ⊊{\mathfrak{p}}_n$ of prime ideals of $R$. The Krull dimension of $R$ is the supremum of all integers $n\ge 0$ for which such a chain exists. This supremum is allowed to be infinite. On this page the zero ring is left outside the definition so that later chain statements do not hide that degenerate boundary. (Krull dimension of a nonzero ring).

Given: The objects and hypotheses in the statement.

Every classical variety is Noetherian and has finitely many irreducible components. Every open or closed subvariety has a finite affine cover. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Classical varieties have finite irreducible decompositions).

Assume the Axiom of Choice. Let $X$ be a classical affine variety. If $U\subseteq X$ is a nonempty affine open subset, then $\mathrm{Frac}(k[U])\cong k(X)$ canonically. Hence any two nonempty affine opens of $X$ have canonically isomorphic function fields. (All nonempty affine opens of an irreducible affine variety have the same function field).

Let $\eta :X⇢Y$ be a dominant rational map between classical affine varieties. Then pullback along any representative induces an injective $k$-algebra homomorphism ${\eta }^{\ast }:k(Y)\hookrightarrow k(X)\mathrm{.}$ This construction is independent of the representative and is functorial under composition of dominant rational maps. (Dominant maps pull back function fields functorially).

Given: The objects and hypotheses in the statement.

For a nonempty affine algebraic set $X$, $\dim X=\dim k[X]$, where the right side is Krull dimension. For this comparison only, extend ring dimension to the zero ring by $\dim (0)=-\mathrm{\infty }$; then the equality also holds for $X=\mathrm{\varnothing }$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Affine geometric dimension equals ring dimension).

For irreducible classical $X$, the fraction fields of all nonempty affine charts identify canonically; denote the resulting field by $k(X)$. A dominant morphism $f:X\to Y$ between irreducible classical varieties induces an injection $f^{\ast }:k(Y)\hookrightarrow k(X)$. Dominant means that the image is dense. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Function fields and dominant pullbacks on general varieties).

For every open cover $T={\bigcup }_{i\in I}{U}_i$ of a Noetherian space, $\dim T={\sup }_i\dim U_i$, with empty supremum $-\mathrm{\infty }$. (Dimension can be computed on an open cover).

Let $k$ be a field, let $A$ be a finite-type $k$-domain, and let $K=\mathrm{Frac}(A)$. Then $\dim A={\mathrm{trdeg}}_{k}K\mathrm{.}$ (Affine-domain dimension equals transcendence degree).

Given: The objects and hypotheses in the statement.

If $X$ is an irreducible classical variety, then $\dim X={\mathrm{trdeg}}_{k}k(X)<\mathrm{\infty }$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Dimension equals transcendence degree).

For irreducible classical $X$, the fraction fields of all nonempty affine charts identify canonically; denote the resulting field by $k(X)$. A dominant morphism $f:X\to Y$ between irreducible classical varieties induces an injection $f^{\ast }:k(Y)\hookrightarrow k(X)$. Dominant means that the image is dense. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Function fields and dominant pullbacks on general varieties).

For a classical variety $X$, let $\dim X$ be its chain dimension. If $X_1,\dots ,X_m$ are its irreducible components and $x\in X$ is a closed point, define ${\dim }_{x}X={\max }_{x\in X_i}\dim X_i$. The indexing family is nonempty. Say that $X$ has pure dimension $d$ if every irreducible component has dimension $d$; the condition on components is vacuous for the empty variety, whose dimension is nevertheless $-\mathrm{\infty }$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Global and local dimension of classical varieties).

Given: The objects and hypotheses in the statement.

If $X$ is an irreducible classical variety, then $\dim X={\mathrm{trdeg}}_{k}k(X)<\mathrm{\infty }$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Dimension equals transcendence degree).

For every open cover $T={\bigcup }_{i\in I}{U}_i$ of a Noetherian space, $\dim T={\sup }_i\dim U_i$, with empty supremum $-\mathrm{\infty }$. (Dimension can be computed on an open cover).

For every $i$, normalization of the $i$th coordinate identifies $D_{+}(x_i)$ with ${\mathbf{A}}_k^n$. In particular $[a_0:\cdots :a_n]\mapsto (a_1\mathrm{/}{a}_0,\dots ,a_n\mathrm{/}{a}_0)$ identifies $D_{+}(x_0)$ with ${\mathbf{A}}_k^n$. (standard projective opens are affine spaces).

Given: The objects and hypotheses in the statement.

If $U$ is a nonempty open of an irreducible classical variety $X$, then $\dim U=\dim X$. Every proper closed subvariety $Z⊊X$ has $\dim Z<\dim X$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Nonempty opens preserve irreducible dimension).

Let $X$ and $Y$ be classical affine varieties. Then $X$ and $Y$ are birationally equivalent if and only if their function fields are isomorphic as extensions of $k$. (Irreducible affine varieties are birational exactly when their function fields are isomorphic).

For irreducible classical $X$, the fraction fields of all nonempty affine charts identify canonically; denote the resulting field by $k(X)$. A dominant morphism $f:X\to Y$ between irreducible classical varieties induces an injection $f^{\ast }:k(Y)\hookrightarrow k(X)$. Dominant means that the image is dense. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Function fields and dominant pullbacks on general varieties).

Given: The objects and hypotheses in the statement.

If $X$ is an irreducible classical variety, then $\dim X={\mathrm{trdeg}}_{k}k(X)<\mathrm{\infty }$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Dimension equals transcendence degree).

Every classical variety is Noetherian and has finitely many irreducible components. Every open or closed subvariety has a finite affine cover. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Classical varieties have finite irreducible decompositions).

If a Noetherian space $T$ is a finite union of closed subsets $T_1,\dots ,T_m$, then $\dim T={\max }_i\dim T_i$. For $m=0$ both sides are $-\mathrm{\infty }$. (Dimension of a finite closed union).

For every open cover $T={\bigcup }_{i\in I}{U}_i$ of a Noetherian space, $\dim T={\sup }_i\dim U_i$, with empty supremum $-\mathrm{\infty }$. (Dimension can be computed on an open cover).

Let $X,Y$ be classical affine varieties over an algebraically closed field $k$. Then their affine product exists, is a classical affine variety, and has coordinate ring $k[X{\times }_{k}Y]\cong k[X]{\otimes }_{k}k[Y]\mathrm{.}$ Its projections make it a product in the classical affine-variety category. (The product of affine varieties has coordinate ring k[X] tensor_k k[Y]).

Fix the page's algebraically closed field $k$. Let $\mathcal{C}$ be the category whose objects are classical affine or projective algebraic sets over $k$ (including empty and reducible ones), and whose arrows are regular $k$-maps. Objects isomorphic to such sets are understood with their transported algebraic structure. No existence of products for arbitrary mixed affine/projective factors is asserted. For $X,Y$ in $\mathcal{C}$, a constructed product $X{\times }_{k}Y$ is an object of $\mathcal{C}$ with morphisms $p:X{\times }_{k}Y\to X$ and $q:X{\times }_{k}Y\to Y$ such that, for every object $T$ of $\mathcal{C}$ and morphisms $f:T\to X$, $g:T\to Y$, there is a unique morphism $⟨f,g⟩:T\to X{\times }_{k}Y$ satisfying $p⟨f,g⟩=f$ and $q⟨f,g⟩=g$. Thus it is the categorical product of def-products-and-coproducts in $\mathcal{C}$. The underlying set is written as pairs when a construction supplies that identification. If either factor is empty, the product set is empty. A product with the one-point affine algebraic set has the evident projection isomorphism. When $X,Y$ are varieties, this definition is used only after a construction shows that the resulting nonempty algebraic set is irreducible. (Products of classical algebraic sets and their universal property).

Let $k$ be a field and let $A$ be a nonzero finite-type $k$-algebra. Then there exist algebraically independent elements $z_1,\dots ,z_d\in A$ such that $A$ is a module-finite algebra over the polynomial ring $k[z_1,\dots ,z_d]$. (Noether normalisation yields module finiteness over a polynomial subring).

Given: The objects and hypotheses in the statement.

For a nonempty affine algebraic set $X$, $\dim X=\dim k[X]$, where the right side is Krull dimension. For this comparison only, extend ring dimension to the zero ring by $\dim (0)=-\mathrm{\infty }$; then the equality also holds for $X=\mathrm{\varnothing }$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Affine geometric dimension equals ring dimension).

For a nonempty irreducible closed subvariety $Z$ of an irreducible classical variety $X$, define ${\mathrm{codim}}_{X}Z=\dim X-\dim Z$. These are finite integers. In a reducible ambient variety a difference of global dimensions must not be substituted for the height of a local prime; the containing component matters. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Codimension of an irreducible closed subvariety).

Let $R$ be a Noetherian commutative ring, let $x\in R$, and let $\mathfrak{p}$ be a prime ideal minimal over $(x)$. Then $\mathrm{ht}(\mathfrak{p})\le 1$. (Krull's principal ideal theorem).

Let $k$ be a field, let $A$ be a finite-type $k$-domain, and let $\mathfrak{p}\in \mathrm{Spec}(A)$. Then $\mathrm{ht}(\mathfrak{p})+\dim (A\mathrm{/}\mathfrak{p})=\dim A\mathrm{.}$ (Height plus quotient dimension equals ambient dimension in an affine domain).

Assume the Axiom of Choice. Let $k$ be an algebraically closed field. 1. The assignments $X⟼I(X),J⟼V(J)$ induce mutually inverse inclusion-reversing correspondences between affine algebraic sets $X\subseteq {\mathbf{A}}_k^n$ and radical ideals $J\subseteq k[x_1,\dots ,x_n]$. 2. Under this correspondence, nonempty irreducible affine algebraic sets correspond exactly to prime ideals. (Affine algebraic sets correspond to radical ideals, and irreducible ones to prime ideals).

Given: The objects and hypotheses in the statement.

For a nonempty affine algebraic set $X$, $\dim X=\dim k[X]$, where the right side is Krull dimension. For this comparison only, extend ring dimension to the zero ring by $\dim (0)=-\mathrm{\infty }$; then the equality also holds for $X=\mathrm{\varnothing }$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Affine geometric dimension equals ring dimension).

If $U$ is a nonempty open of an irreducible classical variety $X$, then $\dim U=\dim X$. Every proper closed subvariety $Z⊊X$ has $\dim Z<\dim X$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Nonempty opens preserve irreducible dimension).

Let $R$ be a Noetherian commutative ring, let $I=(x_1,\dots ,x_n)$ be an ideal generated by $n\ge 1$ elements, and let $\mathfrak{p}$ be a prime ideal minimal over $I$. Then $\mathrm{ht}(\mathfrak{p})\le n$. (Krull's height theorem).

Let $k$ be a field, let $A$ be a finite-type $k$-domain, and let $\mathfrak{p}\in \mathrm{Spec}(A)$. Then $\mathrm{ht}(\mathfrak{p})+\dim (A\mathrm{/}\mathfrak{p})=\dim A\mathrm{.}$ (Height plus quotient dimension equals ambient dimension in an affine domain).

Given: The objects and hypotheses in the statement.

If $X$ is an irreducible classical variety, then $\dim X={\mathrm{trdeg}}_{k}k(X)<\mathrm{\infty }$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Dimension equals transcendence degree).

For a nonempty irreducible closed subvariety $Z$ of an irreducible classical variety $X$, define ${\mathrm{codim}}_{X}Z=\dim X-\dim Z$. These are finite integers. In a reducible ambient variety a difference of global dimensions must not be substituted for the height of a local prime; the containing component matters. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Codimension of an irreducible closed subvariety).

Assume the Axiom of Choice. Let $X$ be a classical affine variety over an algebraically closed field $k$, let $x\in X$, and let ${\mathfrak{m}}_x:=\{\bar{f}\in k[X]:\bar{f}(x)=0\}\mathrm{.}$ Then there is a canonical isomorphism of local rings ${\mathcal{O}}_{X,x}\stackrel{\sim }{\to }k[X{]}_{{\mathfrak{m}}_x}\mathrm{.}$ (The local ring at a point of an affine variety is the localization at its maximal ideal).

Let $k$ be a field, let $A$ be a finite-type $k$-domain, and let $\mathfrak{p}\in \mathrm{Spec}(A)$. Then $\mathrm{ht}(\mathfrak{p})+\dim (A\mathrm{/}\mathfrak{p})=\dim A\mathrm{.}$ (Height plus quotient dimension equals ambient dimension in an affine domain).

Let $R$ be a commutative ring and let $\mathfrak{p}\in \mathrm{Spec}(R)$. The height of $\mathfrak{p}$ is the Krull dimension of the local ring $R_{\mathfrak{p}}$: $\mathrm{ht}(\mathfrak{p})=\dim (R_{\mathfrak{p}})\mathrm{.}$ (The height of a prime ideal).

Given: The objects and hypotheses in the statement.

If $X$ is an irreducible classical variety and $x$ is a closed point, then $\dim {\mathcal{O}}_{X,x}=\dim X={\mathrm{codim}}_X\{x\}$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Closed-point local dimension equals ambient irreducible dimension).

Let $R$ be a Noetherian commutative ring and let $\mathfrak{p}\in \mathrm{Spec}(R)$ have finite height $n$. Then in the local ring $R_{\mathfrak{p}}$ there exist elements $x_1,\dots ,x_n\in \mathfrak{p}$ such that the maximal ideal $\mathfrak{p}{R}_{\mathfrak{p}}$ is minimal over $(x_1\mathrm{/}1,\dots ,x_n\mathrm{/}1)$. Equivalently, $\mathfrak{p}$ is minimal over an $n$-generated ideal after localizing at $\mathfrak{p}$. (Converse to Krull's height theorem in localised form).

Every classical variety is Noetherian and has finitely many irreducible components. Every open or closed subvariety has a finite affine cover. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Classical varieties have finite irreducible decompositions).

Given: The objects and hypotheses in the statement.

For a classical variety $X$, let $\dim X$ be its chain dimension. If $X_1,\dots ,X_m$ are its irreducible components and $x\in X$ is a closed point, define ${\dim }_{x}X={\max }_{x\in X_i}\dim X_i$. The indexing family is nonempty. Say that $X$ has pure dimension $d$ if every irreducible component has dimension $d$; the condition on components is vacuous for the empty variety, whose dimension is nevertheless $-\mathrm{\infty }$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Global and local dimension of classical varieties).

Every classical variety is Noetherian and has finitely many irreducible components. Every open or closed subvariety has a finite affine cover. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Classical varieties have finite irreducible decompositions).

Given: The objects and hypotheses in the statement.

Let $X$ be irreducible affine and $0\mathrm{\ne }f\in k[X]$ be a nonunit. Then $V_X(f)$ is nonempty and every irreducible component has dimension $\dim X-1$, hence codimension one. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (A nontrivial principal section has pure codimension one).

If $U$ is a nonempty open of an irreducible classical variety $X$, then $\dim U=\dim X$. Every proper closed subvariety $Z⊊X$ has $\dim Z<\dim X$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Nonempty opens preserve irreducible dimension).

A classical variety $X$ has $\dim X\le 0$ if and only if its underlying set is finite. The empty set is included. A nonempty irreducible variety of dimension zero is one point. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Zero-dimensional varieties are finite sets).

For a nonempty irreducible closed subvariety $Z$ of an irreducible classical variety $X$, define ${\mathrm{codim}}_{X}Z=\dim X-\dim Z$. These are finite integers. In a reducible ambient variety a difference of global dimensions must not be substituted for the height of a local prime; the containing component matters. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Codimension of an irreducible closed subvariety).

Given: The objects and hypotheses in the statement.

A subset $S$ of a classical variety $X$ is locally closed if $S=U\cap Z$ for some open $U\subseteq X$ and closed $Z\subseteq X$. A subset is constructible if it is a finite union of locally closed subsets. The empty union is allowed, so $\mathrm{\varnothing }$ is constructible. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Locally closed and constructible subsets).

Given: The objects and hypotheses in the statement.

A subset $S$ of a classical variety $X$ is locally closed if $S=U\cap Z$ for some open $U\subseteq X$ and closed $Z\subseteq X$. A subset is constructible if it is a finite union of locally closed subsets. The empty union is allowed, so $\mathrm{\varnothing }$ is constructible. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Locally closed and constructible subsets).

Given: The objects and hypotheses in the statement.

For irreducible classical $X$, the fraction fields of all nonempty affine charts identify canonically; denote the resulting field by $k(X)$. A dominant morphism $f:X\to Y$ between irreducible classical varieties induces an injection $f^{\ast }:k(Y)\hookrightarrow k(X)$. Dominant means that the image is dense. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Function fields and dominant pullbacks on general varieties).

Let $k$ be a field and let $A$ be a nonzero finite-type $k$-algebra. Then there exist algebraically independent elements $z_1,\dots ,z_d\in A$ such that $A$ is a module-finite algebra over the polynomial ring $k[z_1,\dots ,z_d]$. (Noether normalisation yields module finiteness over a polynomial subring).

Assume the Axiom of Choice. Let $X$ be a classical affine variety over an algebraically closed field $k$, and let $\bar{f}\in k[X]$. Put $U=D_X(\bar{f})$. A function $\phi :U\to k$ is called regular on $U$ if there exist finitely many pairs $({\bar{g}}_i,{\bar{h}}_i)$ in $k[X]\times k[X]$ such that $U={\bigcup }_{i=1}^r{D}_X({\bar{h}}_i)$ and $\phi (x)=\frac{{\bar{g}}_i(x)}{{\bar{h}}_i(x)}\text{for every }x\in D_X({\bar{h}}_i)\mathrm{.}$ Write ${\mathcal{O}}_X(U)$ for the ring of regular functions on $U$. Then evaluation induces a ring isomorphism $k[X{]}_{\bar{f}}\stackrel{\sim }{\to }{\mathcal{O}}_X(U)\mathrm{.}$ If $U=\mathrm{\varnothing }$, both sides are the zero ring. (Regular functions on a principal open are the principal localization of the coordinate ring).