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 A=R/I be the algebra of the polynomials 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 is does not change if the xi are replaced by linearly independent linear combinations of them. The dimension of V is
![R=K[x_{1},\ldots ,x_{n}].](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c09cfbe2f8a7e95a74bba0e6d059748bf776feb)
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