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.

Read more about this topic:  Algebraic Variety

Famous quotes containing the words basic and/or results:

    “... the basic experience of everyone is the experience of human limitation.”
    —Flannery O’Connor (1925–1964)

    “Consider what you have in the smallest chosen library. A company of the wisest and wittiest men that could be picked out of all civil countries in a thousand years have set in best order the results of their learning and wisdom. The men themselves were hid and inaccessible, solitary, impatient of interruption, fenced by etiquette; but the thought which they did not uncover in their bosom friend is here written out in transparent words to us, the strangers of another age.”
    —Ralph Waldo Emerson (1803–1882)