Affine algebraic set

affine algebraic $k$-set

The set of solutions of a given system of algebraic equations. Let $k$ be a field and let $\bar k$ be its algebraic closure. A subset $X$ of the Cartesian product ${\bar k}^n$ is said to be an affine algebraic $k$-set if its points are the common zeros of some family $S$ of the ring of polynomials $k[T]=k[T_1,\dots,T_n]$. The set ${\mathfrak A}_X$ of all polynomials in $k[T_1,\dots,T_n]$ that vanish on $X$ forms an ideal, the so-called ideal of the affine algebraic $k$-set. The ideal ${\mathfrak A}_X$ coincides with the radical of the ideal $I(S)$ generated by the family $S$, i.e. with the set of polynomials $f\in k[T_1,\dots,T_n]$ such that $f^m \in I(S)$ for some natural number $m$ (Hilbert's Nullstellensatz; cf. Hilbert theorem 3)). Two affine algebraic sets $X$ and $Y$ coincide if and only if ${\mathfrak A}_X = {\mathfrak A}_Y$. The affine algebraic set $X$ can be defined by a system of generators of ${\mathfrak A}_X$. In particular, any affine algebraic set can be defined by a finite number of polynomials $f_1,\dots,f_k\in k[T]$. The equalities $f_1 = \dots = f_k = 0$ are called the equations of $X$. The affine algebraic sets of ${\bar k}^n$ form a lattice with respect to the operations of intersection and union. The ideal of the intersection $X\cap Y$ is identical with the sum of their ideals ${\mathfrak A}_X + {\mathfrak A}_Y$, while the ideal of the union $X\cup Y$ is identical with the intersection of their ideals ${\mathfrak A}_X \cap {\mathfrak A}_Y$. Any set ${\bar k}^n$ is an affine algebraic set, called an affine space over $k$ and denoted by $A_k^n$; to it corresponds the zero ideal. The empty subset of ${\bar k}^n$ is also an affine algebraic set with the unit ideal. The quotient ring $k[X]=k[T]/{\mathfrak A}_X$ is called the coordinate ring of $X$. It is identical with the ring of $k$-regular functions on $X$, i.e. with the ring of $k$-valued functions, $f:X \to {\bar k}$, for which there exists a polynomial $F\in k[T]$ such that $f(x)=F(x)$ for all $x\in X$. 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 ${\mathfrak A}_X$ 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 $k$ (or $k$-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) MR0389876 MR0384768 Zbl 0313.13001 [2] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 [3] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001