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: In particular, if is an affine scheme, then such chains correspond to chains of prime ideals and 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: An irreducible subset of X is an irreducible component of Xif and only if the codimension of it in X is zero. If 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 spaceV 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, k a field. Then it has dimension 2. If x is a closed point of X, then is 2 if x lies in H and is 1 if it is in. Thus, for closed points x can vary.
Let R be a discrete valuation ring and the affine line over it. Let be the projection. consists of 2 points, corresponding to the maximal ideal and closed and the zero ideal and open. Then the fibers are closed and open, respectively. We note that has dimension one, while has dimension and is dense in. 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 be the maximal ideal of R and a generator. We note that has height-two and height-one maximal ideals; namely, and the kernel of. The first ideal is maximal since the field of fractions of R. Also, has height one by Krull's principal ideal theorem and has height two since. Consequently,
Equidimensional scheme
An equidimensional scheme is a scheme all of whose irreducible components are of the same dimension.
Examples
All irreducible schemes are equidimensional. In affine space, the union of a line and a point not on the line is not equidimensional. In general, 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 over Spec k for some field k, then every connected component, is equidimensional.
