In algebraic geometry, the dimension of a scheme is a generalization of a 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.
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 (inclusion reversed) and so the dimension of X is precisely the Krull dimension of A.
is affine, then the codimension of Y in X is precisely the height of the prime ideal defining Y in X.
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 (for instance, étale) over Spec k for some field k, then every connected component (which is then in fact an irreducible component), is equidimensional.
be a morphism locally of finite type between two schemes
If all the nonempty fibers [clarification needed] are purely of the same dimension