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

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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

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  • If a finite-dimensional vector space V over a field is viewed as a scheme over the field,[note 1] then the dimension of the scheme V is the same as the vector-space dimension of V.
  • Let  , k a field. Then it has dimension 2 (since it contains the hyperplane   as an irreducible component). If x is a closed point of X, then   is 2 if x lies in H and is 1 if it is in  . Thus,   for closed points x can vary.
  • Let   be an algebraic pre-variety; i.e., an integral scheme of finite type over a field  . Then the dimension of   is the transcendence degree of the function field   of   over  .[3] Also, if   is a nonempty open subset of  , then  .[4]
  • Let R be a discrete valuation ring and   the affine line over it. Let   be the projection.   consists of 2 points,   corresponding to the maximal ideal and closed and   the zero ideal and open. Then the fibers   are closed and open, respectively. We note that   has dimension one,[note 2] while   has dimension   and   is dense in  . Thus, the dimension of the closure of an open subset can be strictly bigger than that of the open set.
  • Continuing the same example, let   be the maximal ideal of R and   a generator. We note that   has height-two and height-one maximal ideals; namely,   and   the kernel of  . The first ideal   is maximal since   the field of fractions of R. Also,   has height one by Krull's principal ideal theorem and   has height two since  . Consequently,
 
while X is irreducible.

Equidimensional scheme

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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

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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

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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

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Notes

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  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.

References

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