Dimension of a scheme
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. 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: In particular, if formula_2 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". 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 "X" if and only if the codimension of it in "X" is zero. If formula_2 is affine, then the codimension of "Y" in "X" is precisely the height of the prime ideal defining "Y" in "X". Equidimensional scheme. An equidimensional scheme (or, pure dimensional scheme) is a scheme all of whose irreducible components are of  the same dimension (implicitly assuming the dimensions are all well-defined). 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 (for instance, étale) over Spec "k" for some field "k", then every "connected" component (which is then in fact an irreducible component), is equidimensional.