Affine algebraic set

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

affine algebraic -set

The set of solutions of a given system of algebraic equations. Let be a field and let be its algebraic closure. A subset of the Cartesian product is said to be an affine algebraic -set if its points are the common zeros of some family of the ring of polynomials . The set of all polynomials in that vanish on forms an ideal, the so-called ideal of the affine algebraic -set. The ideal coincides with the radical of the ideal generated by the family , i.e. with the set of polynomials such that for some natural number (Hilbert's Nullstellensatz; cf. Hilbert theorem 3)). Two affine algebraic sets and coincide if and only if . The affine algebraic set can be defined by a system of generators of . In particular, any affine algebraic set can be defined by a finite number of polynomials . The equalities are called the equations of . The affine algebraic sets of form a lattice with respect to the operations of intersection and union. The ideal of the intersection is identical with the sum of their ideals , while the ideal of the union is identical with the intersection of their ideals . Any set is an affine algebraic set, called an affine space over and denoted by ; to it corresponds the zero ideal. The empty subset of is also an affine algebraic set with the unit ideal. The quotient ring is called the coordinate ring of . It is identical with the ring of -regular functions on , i.e. with the ring of -valued functions, , for which there exists a polynomial such that for all . An affine algebraic set is said to be irreducible if it is not the union of two affine algebraic proper subsets. An equivalent definition is that the ideal is prime. Irreducible affine algebraic sets together with projective algebraic sets were the subjects of classical algebraic geometry. They were called, respectively, affine algebraic varieties and projective algebraic varieties over the field (or -varieties). Affine algebraic sets have the structure of a topological space. The affine algebraic subsets are the closed sets of this topology (the Zariski topology). An affine algebraic set is irreducible if and only if it is irreducible as a topological space. Further development of the concept of an affine algebraic set leads to the concepts of an affine variety and an affine scheme.

References

 [1] O. Zariski, P. Samuel, "Commutative algebra" , 2 , Springer (1975) [2] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) [3] R. Hartshorne, "Algebraic geometry" , Springer (1977)