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Zariski and topology
* The spectrum of any commutative ring with the Zariski topology ( that is, the set of all prime ideals ) is compact, but never Hausdorff ( except in trivial cases ).
The analytic techniques used for studying Lie groups must be replaced by techniques from algebraic geometry, where the relatively weak Zariski topology causes many technical complications.
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec ( R ), is the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.
This topology is called the Zariski topology.
A basis for the Zariski topology can be constructed as follows.
For f ∈ R, define D < sub > f </ sub > to be the set of ideals of R not containing f. Then each D < sub > f </ sub > is an open subset of Spec ( R ), and is a basis for the Zariski topology.
Given the space X = Spec ( R ) with the Zariski topology, the structure sheaf O < sub > X </ sub > is defined on the D < sub > f </ sub > by setting Γ ( D < sub > f </ sub >, O < sub > X </ sub >) = R < sub > f </ sub >, the localization of R at the multiplicative system
In some contexts, it is helpful to place other topologies on the set of real numbers, such as the lower limit topology or the Zariski topology.
It follows readily from the definition of the spectrum of a ring, the space of prime ideals of equipped with the Zariski topology, that the Krull dimension of is precisely equal to the irreducible dimension of its spectrum.
In a similar vein, the Zariski topology on A < sup > n </ sup > is defined by taking the zero sets of polynomial functions as a base for the closed sets.
The determinant is a polynomial map, and hence GL ( n, R ) is an open affine subvariety of M < sub > n </ sub >( R ) ( a non-empty open subset of M < sub > n </ sub >( R ) in the Zariski topology ), and therefore
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition.
The more subtle étale topology is a refinement of the Zariski topology discovered by Grothendieck in the 1960s that reflects the geometry more accurately.
In classical algebraic geometry ( that is, the subject prior to the Grothendieck revolution of the late 1950s and 1960s ) the Zariski topology was defined in the following way.
Just as the subject itself was divided into the study of affine and projective varieties ( see the Algebraic variety definitions ) the Zariski topology is defined slightly differently for these two.
This is the Zariski topology on
If X is an affine algebraic set ( irreducible or not ) then the Zariski topology on it is defined simply to be the subspace topology induced by its inclusion into some Equivalently, it can be checked that:
This establishes that the above equation, clearly a generalization of the previous one, defines the Zariski topology on any affine variety.
The projective Zariski topology is defined for projective algebraic sets just as the affine one is defined for affine algebraic sets, by taking the subspace topology.

Zariski and on
( This is a little inaccurate: Deligne did later show that E ∩ E < sup >⊥</ sup > = 0 by using the hard Lefschetz theorem, this requires the Weil conjectures, and the proof of the Weil conjectures really has to use a slightly more complicated argument with E / E ∩ E < sup >⊥</ sup > rather than E .) An argument of Kazhdan and Margulis shows that the image of the monodromy group acting on E, given by the Picard – Lefschetz formula, is Zariski dense in a symplectic group and therefore has the same invariants, which are well known from classical invariant theory.
If X is an algebraic variety carrying the Zariski topology, we can define a locally ringed space by taking O < sub > X </ sub >( U ) to be the ring of rational functions defined on the Zariski-open set U which do not blow up ( become infinite ) within U. The important generalization of this example is that of the spectrum of any commutative ring ; these spectra are also locally ringed spaces.
Now define the sheaf on X by: for each Zariski open subset U,
Zariski wrote a doctoral dissertation in 1924 on a topic in Galois theory.
Zariski himself worked on equisingularity theory.
Some of his major results, Zariski's main theorem and the Zariski theorem on holomorphic functions, were amongst the results generalized and included in the programme of Alexander Grothendieck that ultimately unified algebraic geometry.
He defined the spectrum of a commutative ring as the space of prime ideals with Zariski topology, but augments it with a sheaf of rings: to every Zariski-open set he assigns a commutative ring, thought of as the ring of " polynomial functions " defined on that set.
Since polynomials over a field K are zero on finite sets, or the whole of K, the Zariski topology on K ( considered as affine line ) is the cofinite topology.
On an open dense set they do behave like functions, but the Zariski closures of their graphs are more complex correspondences on the product showing ' blowing up ' and ' blowing down '.
If n is divisible by p this argument breaks down because pth roots of 1 behave strangely over fields of characteristic p. In the Zariski topology the Kummer sequence is not exact on the right, as a non-vanishing function does not usually have an nth root locally for the Zariski topology, so this is one place where the use of the étale topology rather than the Zariski topology is essential.
In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point P on an algebraic variety V ( and more generally ).
* Let X be an affine scheme, F a quasi-coherent sheaf of O < sub > X </ sub >- modules for the Zariski topology on X.
The Italian school liked to reduce the geometry on an algebraic surface to that of linear systems cut out by surfaces in three-space ; Zariski wrote his celebrated book Algebraic Surfaces to try to pull together the methods, involving linear systems with fixed base points.
Given a very ample sheaf L on X and a coherent sheaf F, a theorem of Serre shows that ( the coherent sheaf ) F ⊗ L < sup >⊗ n </ sup > is generated by finitely many global sections for sufficiently large n. This in turn implies that global sections and higher ( Zariski ) cohomology groups

Zariski and algebraic
This formulation is analogous to the construction of the cotangent space to define the Zariski tangent space in algebraic geometry.
An algebraic subgroup of an algebraic group is a Zariski closed subgroup.
Projective varieties are also equipped with the Zariski topology by declaring all algebraic sets to be closed.
Equivalently, an algebraic variety is projective if it can be embedded as a ( Zariski ) closed subvariety of P < sup > n </ sup >.
Chow's theorem says that a subvariety of the projective space is analytic ( closed in the strong sense ) if and only if it is algebraic ( closed in the Zariski topology ).
Oscar Zariski ( born Oscher Zaritsky () April 24, 1899, in Kobrin, Russian Empire ( today Belarus ), died July 4, 1986, Brookline, Massachusetts ) was an American mathematician and one of the most influential algebraic geometers of the 20th century.
In the 1930s, Wolfgang Krull turned things around and took a radical step: start with any commutative ring, consider the set of its prime ideals, turn it into a topological space by introducing the Zariski topology, and study the algebraic geometry of these quite general objects.
In 1944 Oscar Zariski defined an abstract Zariski – Riemann space from the function field of an algebraic variety, for the needs of birational geometry: this is like a direct limit of ordinary varieties ( under ' blowing up '), and the construction, reminiscent of locale theory, used valuation rings as points.
From about 1950 to 1980 there was considerable effort to salvage as much as possible from the wreckage, and convert it into the rigorous algebraic style of algebraic geometry set up by Weil and Zariski.

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