Dimension of an algebraic variety

In algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways. Some of them are of geometric nature, while some other are purely algebraic and rely on commutative algebra. Some are restricted to algebraic varieties while others apply also to any algebraic set.

Let V be an algebraic set defined as the set of the common zeros of an ideal I in a polynomial ring $R=K[x_1, \ldots, x_n]$ over a field K, and let A=R/I be the algebra of the polynomials over V. Then the dimension of V is:

• The maximal length of the chains $V_0\subset V_1\subset \ldots \subset V_d$ of sub varieties (the length of such a chain is the number of $\subset$).
• The Krull dimension of A.
• The maximal Krull dimension of the local rings at the points of V.
• The maximal dimension of the tangent vector spaces at the non singular points of V.
• The maximal length of a regular sequence in A.
• The number of hyperplanes in generic position which are needed to have an intersection with V which is reduced to a finite number of points.
• The difference between n and the maximal length of the regular sequences contained in I.
• The degree of the Hilbert polynomial of A.
• The degree of the denominator of the Hilbert series of A
• If I is a prime ideal (i.e. V is an algebraic variety), the transcendence degree over K of the field of fractions of A.

If V is a projective variety defined by a homogeneous ideal I, then the values for which A or I appear explicitly in previous definitions must be decreased by one.

Definition by the transcendence degree

For an algebraic variety V over a field K, the dimension of V is the transcendence degree over K of the function field K(V) of all rational functions on V, with values in K.

For the function field even to be defined, V here must be an irreducible algebraic set; in which case the function field (for an affine variety) is just the field of fractions of the coordinate ring of V. Using polynomial equations, it is easy to define sets that have 'mixed dimension': a union of a curve and a plane in space, for example. These fail to be irreducible.

References

• Hazewinkel, Michiel, ed. (2001), "Algebraic function", Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=a/a011490