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.

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.

The Zariski topology, as it was later known, is adequate for biregular geometry, where varieties are mapped by polynomial functions.

Zariski was awarded the Steele Prize in 1981, and in the same year the Wolf Prize in Mathematics with Lars Ahlfors.

This was an essentially sound, breakthrough set of insights, recovered in modern complex manifold language by Kunihiko Kodaira in the 1950s, and refined to include mod p phenomena by Zariski, the Shafarevich school and others by around 1960.

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.

Valuation theory, too, was an important technical extension, and was used by Helmut Hasse and Oscar Zariski.

One can also say that the diagonalizable matrices form a dense subset with respect to the Zariski topology: the complement lies inside the set where the discriminant of the characteristic polynomial vanishes, which is a hypersurface.

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.

Projective varieties are also equipped with the Zariski topology by declaring all algebraic sets to be closed.

One also says that a certain property is Zariski-local, if one needs to distinguish between the Zariski topology and other possible topologies, like the étale topology.

The relationship between G and K that drives this connection is traditionally expressed in the terms that the Lie algebra of K is a real form of that of G. In the theory of algebraic groups, the relationship can also be put that K is a dense subset of G, for the Zariski topology.

Zariski and Samuel were sufficiently taken by this construction to pose as a question whether every Dedekind domain arises in such a fashion, i. e., by starting with a PID and taking the integral closure in a finite degree field extension.

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 taught at the Universidade de São Paulo, 1945 – 47, where he worked with Oscar Zariski.

After completing his undergraduate studies at Kyoto University, he received his Ph. D. from Harvard while under the direction of Oscar Zariski.

Then the Zariski tangent space at a point p ∈ X is the collection of K-derivations D: O < sub > X, p </ sub >→ K, where K is the ground field and O < sub > X, p </ sub > is the stalk of O < sub > X </ sub > at p.

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

After spending a year 1946 – 1947 at the University of Illinois, Zariski became professor at Harvard University in 1947 where he remained until his retirement in 1969.

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 ).

In the first case the ( Zariski ) tangent space to C at ( 0, 0 ) is the whole plane, considered as a two-dimensional affine space.

The tangent cone to X at the origin is the Zariski closed subset of k < sup > n </ sup > defined by the ideal in ( I ).

Thus the Zariski tangent space to C at the origin is the whole plane, and has higher dimension than the curve itself ( two versus one ).

This is the case, for example, when looking at the category of sheaves on projective space in the Zariski topology.

He earned his undergraduate and graduate degrees at Harvard University, where he earned his Ph. D. in 1950 under Oscar Zariski, introducing in his dissertation Gorenstein rings.

in 1955 ; he then moved to Harvard University, where he received a Ph. D. in 1960 under the supervision of Oscar Zariski.

* 1899 – Oscar Zariski, Russian mathematician ( d. 1986 )

This formulation is analogous to the construction of the cotangent space to define the Zariski tangent space in algebraic geometry.

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.

* 1986 – Oscar Zariski, Russian mathematician ( b. 1899 )

See Zariski tangent space.

* July 4 – Oscar Zariski, Russian mathematician ( b. 1899 )

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.

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.

In fact, more generally, one has a Galois connection between subsets of the space and subsets of the algebra, where " Zariski closure " and " radical of the ideal generated " are the closure operators.

The English term local ring is due to Zariski.

Another higher-dimensional generalization of Faltings ' theorem is the Bombieri – Lang conjecture that if X is a pseudo-canonical variety ( i. e., variety of general type ) over a number field k, then X ( k ) is not Zariski dense in X.