View source for Affine algebraic set

''affine algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a0109402.png" />-set''

The set of solutions of a given system of algebraic equations. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a0109403.png" /> be a field and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a0109404.png" /> be its algebraic closure. A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a0109405.png" /> of the Cartesian product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a0109406.png" /> is said to be an affine algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a0109408.png" />-set if its points are the common zeros of some family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a0109409.png" /> of the [[Ring of polynomials|ring of polynomials]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094010.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094011.png" /> of all polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094012.png" /> that vanish on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094013.png" /> forms an ideal, the so-called ideal of the affine algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094015.png" />-set. The ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094016.png" /> coincides with the radical of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094017.png" /> generated by the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094018.png" />, i.e. with the set of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094019.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094020.png" /> for some natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094021.png" /> (Hilbert's Nullstellensatz; cf. [[Hilbert theorem|Hilbert theorem]] 3)). Two affine algebraic sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094023.png" /> coincide if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094024.png" />. The affine algebraic set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094025.png" /> can be defined by a system of generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094026.png" />. In particular, any affine algebraic set can be defined by a finite number of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094027.png" />. The equalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094028.png" /> are called the equations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094029.png" />. The affine algebraic sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094030.png" /> form a lattice with respect to the operations of intersection and union. The ideal of the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094031.png" /> is identical with the sum of their ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094032.png" />, while the ideal of the union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094033.png" /> is identical with the intersection of their ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094034.png" />. Any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094035.png" /> is an affine algebraic set, called an affine space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094036.png" /> and denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094037.png" />; to it corresponds the zero ideal. The empty subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094038.png" /> is also an affine algebraic set with the unit ideal. The quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094039.png" /> is called the coordinate ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094040.png" />. It is identical with the ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094041.png" />-regular functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094042.png" />, i.e. with the ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094043.png" />-valued functions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094044.png" />, for which there exists a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094045.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094046.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094047.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094048.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094049.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010940/a01094050.png" />-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|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|affine variety]] and an [[Affine scheme|affine scheme]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Zariski,   P. Samuel,   "Commutative algebra" , '''2''' , Springer  (1975)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.R. Shafarevich,   "Basic algebraic geometry" , Springer  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Hartshorne,   "Algebraic geometry" , Springer  (1977)</TD></TR></table>



====Comments====
A topological space is irreducible if it is not the union of two closed proper subspaces.