Zariski topology

The Zariski topology is a term from the mathematical branch of algebraic geometry . It is the natural topology on the objects of study of algebraic geometry, the algebraic varieties or, more generally, the schemes .

The Zariski topology in classical algebraic geometry

In classical algebraic geometry, the Zariski topology (according to Oscar Zariski ) is the topology on the affine space over an algebraically closed body that is derived from the open sets of the form ${\displaystyle K^{n}}$ ${\displaystyle K}$

${\displaystyle \operatorname {D} (f)=\{x\in K^{n}\mid f(x)\neq 0\}}$ For ${\displaystyle f\in K[X_{1},\dotsc ,X_{n}]}$

is produced. Affine varieties carry the induced topology , and the Zariski topology on more general varieties is defined by affine maps .

For example, the Zariski topology on the affine line is the topology of the finite sets.

On an affine variety, the Zariski topology is the coarsest topology for which the regular functions as mappings into the affine straight line (with their Zariski topology) are continuous. ${\displaystyle K}$

The Zariski topology on the spectrum of a ring

If a commutative ring with one element , then the spectrum is the set of the prime ideals of with the topology, in which the closed sets are the sets ${\displaystyle A}$ ${\displaystyle \operatorname {Spec} A}$ ${\displaystyle A}$

${\displaystyle \{{\mathfrak {p}}\in \operatorname {Spec} A\mid {\mathfrak {p}}\supseteq I\}}$

are for ideals . ${\displaystyle I\subseteq A}$

If is for an algebraically closed field , the maximum ideals of correspond to the elements of according to Hilbert's theorem of zeros , and the topologies on these two sets agree. ${\displaystyle A=K[X_{1},\dotsc ,X_{n}]}$ ${\displaystyle K}$ ${\displaystyle A}$ ${\displaystyle K^{n}}$

properties

The Zariski topology is very different from the usual topological spaces based on real numbers.

• The topology is i. A. not Hausdorffsch ; in fact, the space is irreducible , d. That is, every two non-empty open subsets intersect. So irreducibility is a stronger term than context . ${\displaystyle K^{n}}$
• Quasi-compact subsets do not necessarily have to be closed.

Generalizations

• The Zariski topology of a schema is part of its structure; however, the term “Zariski topology” is usually only used in the context of schemes to distinguish it from other Grothendieck topologies .

Individual evidence

1. ↑ Kunz: Introduction to Algebraic Geometry. 1997, p. 26 in the Google book search
2. ↑ Kunz: Introduction to Algebraic Geometry. 1997, p. 66 in the Google book search

literature

• Ernst Kunz : Introduction to Algebraic Geometry. Vieweg, Braunschweig / Wiesbaden 1997, ISBN 978-3-528-07287-2 , p. 26, p. 66.