Affine algebraic set
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) |
Comments
A topological space is irreducible if it is not the union of two closed proper subspaces.
Affine algebraic set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affine_algebraic_set&oldid=13073