Dimension of a scheme

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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.

Contents

Definition

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:

[1]

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.

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:

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

Examples

while X is irreducible.

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. [5]

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.

Relative dimension

Let be a morphism locally of finite type between two schemes and . The relative dimension of at a point is the dimension of the fiber . If all the nonempty fibers [ clarification needed ] are purely of the same dimension , then one says that is of relative dimension . [6]

See also

Notes

  1. The Spec of the symmetric algebra of the dual vector space of V is the scheme structure on .
  2. In fact, by definition, is the fiber product of and and so it is the Spec of .
  1. Hartshorne 1977 , Ch. I, just after Corollary 1.6.
  2. Hartshorne 1977 , Ch. II, just after Example 3.2.6.
  3. Hartshorne 1977 , Ch. II, Exercise 3.20. (b)
  4. Hartshorne 1977 , Ch. II, Exercise 3.20. (e)
  5. Dundas, Bjorn Ian; Jahren, Björn; Levine, Marc; Østvær, P.A.; Röndigs, Oliver; Voevodsky, Vladimir (2007), Motivic Homotopy Theory: Lectures at a Summer School in Nordfjordeid, Norway, August 2002, Springer, p. 101, ISBN   9783540458975 .
  6. Adeel, Ahmed Kahn (March 2013). "Relative Dimension in Ncatlab". Ncatlab . Retrieved 8 June 2022.

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References