Dimension of an algebraic variety

In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways. Some of these definitions 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. Some are intrinsic, as independent of any embedding of the variety into an affine or projective space, while other are related to such an embedding. Let K be a field, and L ⊇ K be an algebraically closed extension. An affine algebraic set V is the set of the common zeros in Ln of the elements of an ideal I in a polynomial ring Let be the algebra of the polynomial functions over V. The dimension of V is any of the following integers. It does not change if K is enlarged, if L is replaced by another algebraically closed extension of K and if I is replaced by another ideal having the same zeros (that is having the same radical). The dimension is also independent of the choice of coordinates; in other words it does not change if the xi are replaced by linearly independent linear combinations of them. The dimension of V is The maximal length of the chains of distinct nonempty (irreducible) subvarieties of V. This definition generalizes a property of the dimension of a Euclidean space or a vector space. It is thus probably the definition that gives the easiest intuitive description of the notion. The Krull dimension of the coordinate ring A. This is the transcription of the preceding definition in the language of commutative algebra, the Krull dimension being the maximal length of the chains of prime ideals of A. The maximal Krull dimension of the local rings at the points of V. This definition shows that the dimension is a local property if is irreducible. If is irreducible, it turns out that all the local rings at closed points have the same Krull dimension (see ). If V is a variety, the Krull dimension of the local ring at any point of V This rephrases the previous definition into a more geometric language.