Dimension of a scheme

In algebraic geometry, the dimension of a scheme is a generalization of the dimension of an algebraic variety. Scheme theory emphasizes the relative point of view and, accordingly, the relative dimension of a morphism of schemes is also important.

Definition

By definition, the dimension of a scheme X is the dimension of the underlying topological space: the supremum of the lengths ℓ of chains of irreducible closed subsets:

$\mathrm{\varnothing }\ne V_0⊊V_1⊊\cdots ⊊V_{\ell }\subset X.$

In particular, if $X=\mathrm{Spec}A$ is an affine scheme, then such chains correspond to chains of prime ideals (inclusion reversed), so the dimension of X is precisely the Krull dimension of A.

If Y is an irreducible closed subset of a scheme X, then the codimension of Y in X is the supremum of the lengths ℓ of chains of irreducible closed subsets:

$Y=V_0⊊V_1⊊\cdots ⊊V_{\ell }\subset X.$

An irreducible subset of X is an irreducible component of X if and only if its codimension in X is zero. If $X=\mathrm{Spec}A$ is affine, then the codimension of Y in X is precisely the height of the prime ideal defining Y in X.

Examples

• If a finite-dimensional vector space V over a field is viewed as a scheme over the field, then the dimension of the scheme V is the same as the vector-space dimension of V.
• Let $X=\mathrm{Spec}k[x,y,z]/(xy,xz)$, k a field. Then it has dimension 2 (since it contains the hyperplane $H=\{x=0\}\subset {𝔸}^3$ as an irreducible component). If x is a closed point of X, then $\mathrm{codim}(x,X)$ is 2 if x lies in H and is 1 if it is in $X-H$. Thus, $\mathrm{codim}(x,X)$ for closed points x can vary.
• Let $X$ be an algebraic pre-variety; i.e., an integral scheme of finite type over a field $k$. Then the dimension of $X$ is the transcendence degree of the function field $k(X)$ of $X$ over $k$. Also, if $U$ is a nonempty open subset of $X$, then $\dim U=\dim X$.
• Let R be a discrete valuation ring and $X={𝔸}_R^1=\mathrm{Spec}(R[t])$ the affine line over it. Let $\pi :X\to \mathrm{Spec}R$ be the projection. $\mathrm{Spec}(R)=\{s,\eta \}$ consists of 2 points, $s$ corresponding to the maximal ideal and closed and $\eta$ the zero ideal and open. Then the fibers ${\pi }^{-1}(s),{\pi }^{-1}(\eta )$ are closed and open, respectively. We note that ${\pi }^{-1}(\eta )$ has dimension one, while $X$ has dimension $2=1+\dim R$ and ${\pi }^{-1}(\eta )$ is dense in $X$. Thus, the dimension of the closure of an open subset can be strictly bigger than that of the open set.
• Continuing the same example, let ${𝔪}_R$ be the maximal ideal of R and ${\omega }_R$ a generator. We note that $R[t]$ has height-two and height-one maximal ideals; namely, ${𝔭}_1=({\omega }_{R}t-1)$ and ${𝔭}_2=$ the kernel of $R[t]\to R/{𝔪}_R,f\mapsto f(0){mod𝔪}_R$. The first ideal ${𝔭}_1$ is maximal since $R[t]/({\omega }_{R}t-1)=R[{\omega }_R^{-1}]=$ the field of fractions of R. Also, ${𝔭}_1$ has height one by Krull's principal ideal theorem and ${𝔭}_2$ has height two since ${𝔪}_R[t]⊊{𝔭}_2$. Consequently,

$\mathrm{codim}({𝔭}_1,X)=1,\mathrm{codim}({𝔭}_2,X)=2,$

while X is irreducible.

Equidimensional scheme

An equidimensional scheme (or, pure dimensional scheme) is a scheme whose irreducible components are of the same dimension (implicitly assuming the dimensions are all well-defined).

Examples

All irreducible schemes are equidimensional.

In an affine space, the union of a line and a point not on the line is not equidimensional. Generally, if two closed subschemes of some scheme, neither containing the other, have unequal dimensions, then their union is not equidimensional.

If a scheme is smooth (for instance, étale) over Spec k for some field k, then every connected component (which is then, in fact, an irreducible component) is equidimensional.

Relative dimension

Let $f:X\to Y$ be a morphism locally of finite type between two schemes $X$ and $Y$. The relative dimension of $f$ at a point $y\in Y$ is the dimension of the fiber $f^{-1}(y)$. If all the nonempty fibers are purely of the same dimension $n$, then one says that $f$ is of relative dimension $n$.

See also

• Kleiman's theorem
• Glossary of scheme theory
• Equidimensional ring