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:

∅︀ \neq V_{0} ⊊ V_{1} ⊊ \dots ⊊ V_{ℓ} \subset X .

In particular, if $X = Spec A$ 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} ⊊ \dots ⊊ V_{ℓ} \subset X .

An irreducible subset of X is an irreducible component of X if and only if its codimension in X is zero. If $X = Spec A$ 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 $X = Spec k [x, y, z] / (x y, x z)$ , 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 $codim (x, X)$ is 2 if x lies in H and is 1 if it is in $X - H$ . Thus, $codim (x, X)$ 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 $X = 𝔸_{R}^{1} = Spec (R [t])$ the affine line over it. Let $π : X \to Spec R$ be the projection. $Spec (R) = {s, η}$ consists of 2 points, $s$ corresponding to the maximal ideal and closed and $η$ the zero ideal and open. Then the fibers $π^{- 1} (s), π^{- 1} (η)$ are closed and open, respectively. We note that $π^{- 1} (η)$ has dimension one, while $X$ has dimension $2 = 1 + \dim R$ and $π^{- 1} (η)$ 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 $ω_{R}$ a generator. We note that $R [t]$ has height-two and height-one maximal ideals; namely, $𝔭_{1} = (ω_{R} t - 1)$ and $𝔭_{2} =$ the kernel of $R [t] \to R / 𝔪_{R}, f \mapsto f (0) {mod 𝔪}_{R}$ . The first ideal $𝔭_{1}$ is maximal since $R [t] / (ω_{R} t - 1) = R [ω_{R}^{- 1}] =$ 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,

codim (𝔭_{1}, X) = 1, 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 ^{[clarification needed]} are purely of the same dimension

n

, then one says that

f

is of relative dimension

n

.

Dimension of a scheme

Definition

Examples

Equidimensional scheme

Examples

Relative dimension

See also