Algebraic Variety - Basic Results

Basic Results

  • An affine algebraic set V is a variety if and only if I(V) is a prime ideal; equivalently, V is a variety if and only if its coordinate ring is an integral domain.
  • Every nonempty affine algebraic set may be written uniquely as a union of algebraic varieties (where none of the sets in the decomposition are subsets of each other).
  • Let k be the coordinate ring of the variety V. Then the dimension of V is the transcendence degree of the field of fractions of k over k.

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