Abstract variety
Encyclopedia
In mathematics
, in the field of algebraic geometry
, the idea of abstract variety is to define a concept of algebraic variety
in an intrinsic way. This followed the trend in the definition of manifold
independent of any ambient space (Hassler Whitney
, in the 1930s) by some years, the first notions being those of Oscar Zariski
and André Weil
in the 1940s. It was Weil, in his foundational work, who gave a first acceptable definition of algebraic variety that stood outside projective space
.
The simplest notion of algebraic variety is affine algebraic variety. If k is a given algebraically closed field, then An(k) is the n-fold Cartesian product of k with itself. Given an ideal I in the ring k[x1,...,xn] of polynomials in n variables over k, the zero set V(I) is the affine variety defined by the ideal. Unfortunately, affine varieties lack a fundamental property known as completeness. To address this deficiency, affine varieties can be completed, by embedding them in projective space. Formally, a new variable x0 is introduced, and the polynomials are replaced by homogeneous polynomials. Choosing an index i to omit from the defining polynomials provide an affine subspace of Pn, and an open affine subvariety of the projective variety. The problem with this approach is that the mechanics of working with projective space and homogeneous coordinates is not terribly geometrical, and is also somewhat arbitrary. Taking a step back, we see that if V is projective variety, the set of affine varieties we have defined is an open cover of V. Moreover, if Uα is an element of the open cover, there is an associated affine coordinate ring O(Uα), and the assignment of coordinate rings to these sets forms a presheaf on V which will be known as the structure sheaf. An abstract variety (V,O) is a topological space V with an associated sheaf O of commutative rings that has the additional property that it can be covered by open sets U such that (V|U, OU) is isomorphic to an affine variety, and such that any morphism of abstract varieties corresponds to morphisms of affine varieties in a neighborhood of any point. Just as in the classical case, the topology of V is known as the Zariski topology
.
. The difference is that schemes are not inherently tied to An or to rings of polynomials. Instead, the topology and structure sheaf are defined directly from the ideal structure of commutative rings.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the field of algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, the idea of abstract variety is to define a concept of algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
in an intrinsic way. This followed the trend in the definition of manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
independent of any ambient space (Hassler Whitney
Hassler Whitney
Hassler Whitney was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.-Work:...
, in the 1930s) by some years, the first notions being those of Oscar Zariski
Oscar Zariski
Oscar Zariski was a Russian mathematician and one of the most influential algebraic geometers of the 20th century.-Education:...
and André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
in the 1940s. It was Weil, in his foundational work, who gave a first acceptable definition of algebraic variety that stood outside projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
.
The simplest notion of algebraic variety is affine algebraic variety. If k is a given algebraically closed field, then An(k) is the n-fold Cartesian product of k with itself. Given an ideal I in the ring k[x1,...,xn] of polynomials in n variables over k, the zero set V(I) is the affine variety defined by the ideal. Unfortunately, affine varieties lack a fundamental property known as completeness. To address this deficiency, affine varieties can be completed, by embedding them in projective space. Formally, a new variable x0 is introduced, and the polynomials are replaced by homogeneous polynomials. Choosing an index i to omit from the defining polynomials provide an affine subspace of Pn, and an open affine subvariety of the projective variety. The problem with this approach is that the mechanics of working with projective space and homogeneous coordinates is not terribly geometrical, and is also somewhat arbitrary. Taking a step back, we see that if V is projective variety, the set of affine varieties we have defined is an open cover of V. Moreover, if Uα is an element of the open cover, there is an associated affine coordinate ring O(Uα), and the assignment of coordinate rings to these sets forms a presheaf on V which will be known as the structure sheaf. An abstract variety (V,O) is a topological space V with an associated sheaf O of commutative rings that has the additional property that it can be covered by open sets U such that (V|U, OU) is isomorphic to an affine variety, and such that any morphism of abstract varieties corresponds to morphisms of affine varieties in a neighborhood of any point. Just as in the classical case, the topology of V is known as the Zariski topology
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
.
Relationship with schemes
The notion of abstract variety is closely analogous to that of a schemeScheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
. The difference is that schemes are not inherently tied to An or to rings of polynomials. Instead, the topology and structure sheaf are defined directly from the ideal structure of commutative rings.