Algebraic set

In mathematics , more precisely in algebraic geometry , an algebraic set is a structure in the plane, in space, or more generally in -dimensional space, which is given by one or more polynomial equations . That is, an algebraic set is the solution set of a system of polynomial equations. ${\displaystyle n}$

In three-dimensional space, for example, the circle in the plane with the center and radius 2 is an algebraic set, because it is the set of solutions for the two equations and . ${\displaystyle z=1}$ ${\displaystyle (0,0,1)}$ ${\displaystyle x^{2}+y^{2}=4}$ ${\displaystyle z=1}$

In older sources and also in some modern introductions, algebraic sets are also called varieties . According to modern usage, only the irreducible algebraic sets count as varieties .

definition

Let be a field and be elements of the polynomial ring in indefinite. The vanishing set of these polynomials is then the subset of given by ${\displaystyle k}$ ${\displaystyle f_{1},f_{2},\ldots ,f_{s}}$ ${\displaystyle k[X_{1},X_{2},\ldots ,X_{n}]}$ ${\displaystyle n}$ ${\displaystyle V(f_{1},f_{2},\ldots ,f_{s})}$ ${\displaystyle k^{n}}$

${\displaystyle V(f_{1},f_{2},\ldots ,f_{s})=\{P\in k^{n}\mid f_{1}(P)=f_{2}(P)=\cdots =f_{s}(P)=0\}\,.}$

A subset is called an affine algebraic set if there are polynomials such that . ${\displaystyle V\subseteq k^{n}}$ ${\displaystyle f_{1},f_{2},\ldots ,f_{s}}$ ${\displaystyle V=V(f_{1},f_{2},\ldots ,f_{s})}$

For example, the parabola is the algebraic set . ${\displaystyle y=x^{2}}$ ${\displaystyle V(Y-X^{2})\subseteq \mathbb {R} ^{2}}$

More generally, if a set of polynomials is made up , one sets . Now be the ideal generated by . One then shows that it holds. According to Hilbert's basic theorem , the ideal is again generated by finitely many polynomials . Thus applies . This means that every algebraic set can be described by a finite number of polynomials. ${\displaystyle T}$ ${\displaystyle k[X_{1},X_{2},\ldots ,X_{n}]}$ ${\displaystyle V(T)=\{P\in k^{n}\mid \forall \,f\in T:f(P)=0\}}$ ${\displaystyle I\trianglelefteq k[X_{1},X_{2},\ldots ,X_{n}]}$ ${\displaystyle T}$ ${\displaystyle V(T)=V(I)}$ ${\displaystyle I}$ ${\displaystyle g_{1},g_{2},\ldots ,g_{t}\in k[X_{1},X_{2},\ldots ,X_{n}]}$ ${\displaystyle V(T)=V(I)=V(g_{1},g_{2},\ldots ,g_{t})}$

Irreducibility

An algebraic set is called irreducible if it cannot be broken down into simpler parts. More precisely, an algebraic set is irreducible if it is not empty and for every pair of algebraic sets with ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle U,W\subset V}$

${\displaystyle U\cup W=V}$

holds that or is. ${\displaystyle U=V}$ ${\displaystyle W=V}$

In other words: is an irreducible algebraic set if is irreducible with respect to the Zariski topology . ${\displaystyle V}$ ${\displaystyle V}$

For example, the union of the -axis and the -axis . Thus is reducible. ${\displaystyle V(XY)\subseteq \mathbb {R} ^{2}}$ ${\displaystyle x}$ ${\displaystyle V(Y)}$ ${\displaystyle y}$ ${\displaystyle V(X)}$ ${\displaystyle XY}$

Disappearance deal

If an algebraic set, then its vanishing deal is defined as ${\displaystyle V\subset k^{n}}$

${\displaystyle I(V):=\{f(X_{1},\ldots ,X_{n})\in k\left[X_{1},\ldots ,X_{n}\right]\mid \forall (\alpha _{1},\ldots ,\alpha _{n})\in V:f(\alpha _{1},\ldots ,\alpha _{n})=0\}}$.

${\displaystyle I(V)}$is a radical ideal, so it applies . ${\displaystyle {\sqrt {I(V)}}=I(V)}$

Prime ideals

Let us now assume that the field is algebraically closed . It then turns out that an algebraic set is irreducible if and only if its vanishing ideal is a prime ideal of the polynomial ring. Furthermore, the depiction of the radical ideals on varieties is given by bijective. The reverse mapping is given by . The images exchange quantity inclusions; maximum ideals correspond exactly to the points of the . This is a consequence of Hilbert's zero theorem . ${\displaystyle k}$ ${\displaystyle J\to V(J)}$ ${\displaystyle V\to I(V)}$ ${\displaystyle k^{n}}$

In the case of a main ideal generated by a polynomial , a prime ideal is precisely if it is an irreducible polynomial , i.e. it cannot be decomposed as a product of non-constant factors. ${\displaystyle P\in K\left[x_{1},\ldots ,x_{n}\right]}$ ${\displaystyle I(X)=(P)}$ ${\displaystyle (P)}$ ${\displaystyle P}$

Decomposition of a variety into irreducible components

Every algebraic set can be represented in a unique way as a finite union of irreducible sub-varieties with for . ${\displaystyle X_{i}}$ ${\displaystyle X_{i}\not \subset X_{j}}$ ${\displaystyle i\not =j}$

Examples

• If there is a regular mapping between projective algebraic sets, and if irreducible and all archetypes are irreducible of the same dimension, then is irreducible. ${\displaystyle \pi \colon X\to Y}$ ${\displaystyle Y}$ ${\displaystyle \pi ^{-1}(p)}$ ${\displaystyle X}$
• If a variety and its universal hyperplane intersection is, then is irreducible. ${\displaystyle X\subset P^{n}}$ ${\displaystyle \Omega _{X}\subset P^{n*}\times X}$ ${\displaystyle \Omega _{X}}$

literature

• Harris, Joe: Algebraic geometry. A first course. Corrected reprint of the 1992 original. Graduate Texts in Mathematics, 133. Springer-Verlag, New York, 1995. ISBN 0-387-97716-3
• Klaus Hulek : Elementary Algebraic Geometry . Basic terms and techniques with numerous examples and applications. 2nd, revised edition. Springer Spectrum, 2012, ISBN 978-3-8348-1964-2 .
• David Cox, John Little, Donal O'Shea: Ideals, Varieties and Algorithms . An Introduction to Computational Algebraic Geometry and Commutative Algebra. 3. Edition. Springer, 2007, ISBN 978-0-387-35650-1 .
• Joachim Hilgert: Mathematical Structures . From linear algebra to rings to geometry with sheaves. Springer Spectrum, Berlin 2016, ISBN 978-3-662-48869-0 .

Individual evidence

1. ^ Jean Dieudonné : The historical development of algebraic geometry . In: American Mathematical Monthly . tape 97 , no. October 8 , 1972, p. 827-866 , p. 838 , JSTOR : 2317664 : "... it was for the first time possible to give a precise meaning to the concepts of dimension and of irreducible variety "
2. ↑ Hulek, p. 20; Cox-Little-O'Shea, p. 5
3. ↑ Robin Hartshorne : Algebraic Geometry (=  Graduate Texts in Mathematics . No. 52 ). Springer, 1977, ISBN 1-4419-2807-3 , p. 3 . David Mumford : The Red Book of Varieties and Schemes (=  Lecture Notes in Mathematics . No.  1358 ). Springer, Berlin 1999, ISBN 3-540-63293-X , p. 30 .
4. ↑ Hilgert, p. 238; Cox-Little-O'Shea p. 5
5. ^ Oprea: Irreducibility and Dimension
6. ^ Harris, Theorem 5.7
7. ↑ Harris, Theorem 11.14
8. ^ Harris, Theorem 5.8