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Dimension of a scheme

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Source updated August 30, 2026

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.

01Definition

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:

\emptyset \neq V_{0}\subsetneq V_{1}\subsetneq \cdots \subsetneq V_{\ell }\subset X.

In particular, if X=\operatorname {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}\subsetneq V_{1}\subsetneq \cdots \subsetneq 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=\operatorname {Spec} A is affine, then the codimension of Y in X is precisely the height of the prime ideal defining Y in X.

02Examples

  • 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=\operatorname {Spec} k[x,y,z]/(xy,xz), k a field. Then it has dimension 2 (since it contains the hyperplane H=\{x=0\}\subset \mathbb {A} ^{3} as an irreducible component). If x is a closed point of X, then \operatorname {codim} (x,X) is 2 if x lies in H and is 1 if it is in X-H. Thus, \operatorname {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=\mathbb {A} _{R}^{1}=\operatorname {Spec} (R[t]) the affine line over it. Let \pi :X\to \operatorname {Spec} R be the projection. \operatorname {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 {\mathfrak {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, {\mathfrak {p}}_{1}=(\omega _{R}t-1) and {\mathfrak {p}}_{2}= the kernel of R[t]\to R/{\mathfrak {m}}_{R},f\mapsto f(0){\bmod {\mathfrak {m}}}_{R}. The first ideal {\mathfrak {p}}_{1} is maximal since R[t]/(\omega _{R}t-1)=R[\omega _{R}^{-1}]= the field of fractions of R. Also, {\mathfrak {p}}_{1} has height one by Krull's principal ideal theorem and {\mathfrak {p}}_{2} has height two since {\mathfrak {m}}_{R}[t]\subsetneq {\mathfrak {p}}_{2}. Consequently,
\operatorname {codim} ({\mathfrak {p}}_{1},X)=1,\,\operatorname {codim} ({\mathfrak {p}}_{2},X)=2,
while X is irreducible.

03Equidimensional 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.

04Relative dimension

Let f:X\rightarrow 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.

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This article is adapted from the Wikipedia article “Dimension of a scheme”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Kleiman's theorem

In algebraic geometry, Kleiman's theorem, introduced by Kleiman, concerns dimension and smoothness of scheme-theoretic intersection after some perturbation of factors in the intersection. Precisely, it states: given a connected algebraic group G acting transitively on an algebraic variety X over an algebraically closed field k and V i → X , i = 1 , 2 {\displaystyle V_{i}\to X,i=1,2} morphisms of varieties, G contains a nonempty open subset such that for each g in the set, either g V 1 × X V 2 {\displaystyle gV_{1}\times _{X}V_{2}} is empty or has pure dimension dim ⁡ V 1 + dim ⁡ V 2 − dim ⁡ X {\displaystyle \dim V_{1}+\dim V_{2}-\dim X} , where g V 1 {\displaystyle gV_{1}} is V 1 → X → g X {\displaystyle V_{1}\to X{\overset {g}{\to }}X} , (Kleiman-Bertini theorem) If V i {\displaystyle V_{i}} are smooth varieties and if the characteristic of the base field k is zero, then g V 1 × X V 2 {\displaystyle gV_{1}\times _{X}V_{2}} is smooth.

Glossary of algebraic geometry

This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory.

Minimal prime ideal

In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal prime ideals.