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.

## . . . Dimension of a scheme . . .

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

${displaystyle X=operatorname {Spec} A}$ 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

is affine, then the codimension of *Y* in *X* is precisely the height of the prime ideal defining *Y* in *X*.

## . . . Dimension of a scheme . . .

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