Abstract variety

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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,...,x_n] of poynomials 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 homogenous 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 working with projective space and homogenous coordinates is not terribly geometrical, and is also somewhat aribtrary. Taking a set 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 U_α is an element of the open cover, there is an associated affine coordinate ring O(U_α), 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, O_U) is isomorphic to an affine variety, and such that any morphism of abstract varieties correspsonds 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.

[edit] Relationship with schemes

The notion of abstract variety is closely analogous to that of a scheme. 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.