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Dimension of a scheme

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Jan 8, 2022

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 of chains of irreducible closed subsets:

V0V1VX.{displaystyle emptyset neq V_{0}subsetneq V_{1}subsetneq cdots subsetneq V_{ell }subset X.}

[1]

In particular, if

X=SpecA{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 of chains of irreducible closed subsets:

Y=V0V1VX.{displaystyle Y=V_{0}subsetneq V_{1}subsetneq cdots subsetneq V_{ell }subset X.}

[2]

An irreducible subset of X is an irreducible component of X if and only if the codimension of it in X is zero. If

X=SpecA{displaystyle 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.

. . . Dimension of a scheme . . .

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. . . Dimension of a scheme . . .