Affine variety
In some contexts (see, for example, Hilbert's Nullstellensatz), it is useful to distinguish the field k in which the coefficients are considered, from the algebraically closed field K (containing k) over which the common zeros are considered (that is, the points of the affine algebraic set are in Kn).
For example, Fermat's Last Theorem asserts that the affine algebraic variety (it is a curve) defined by xn + yn − 1 = 0 has no rational points for any integer n greater than two.
If X is an affine algebraic set, and I is the ideal of all polynomials that are zero on X, then the quotient ring
If X is an affine variety, then I is prime, so the coordinate ring is an integral domain.
They form the ring of regular functions on the variety, or, simply, the ring of the variety; in other words (see #Structure sheaf), it is the space of global sections of the structure sheaf of X.
If V is an affine variety in C2 defined over the complex numbers C, the R-rational points of V can be drawn on a piece of paper or by graphing software.
be a point of V. The Jacobian matrix JV(a) of V at a is the matrix of the partial derivatives The point a is regular if the rank of JV(a) equals the codimension of V, and singular otherwise.
defined by the linear equations[2] If the point is singular, the affine subspace defined by these equations is also called a tangent space by some authors, while other authors say that there is no tangent space at a singular point.
[3] A more intrinsic definition, which does not use coordinates is given by Zariski tangent space.
If k is Noetherian (for instance, if k is a field or a principal ideal domain), then every ideal of k is finitely-generated, so every open set is a finite union of basic open sets.
denote the radical of the ideal I, the set of polynomials f for which some power of f is in I.
The reason that the base field is required to be algebraically closed is that affine varieties automatically satisfy Hilbert's nullstellensatz: for an ideal J in
Hence the correspondence between affine algebraic sets and radical ideals is a bijection.
Prime ideals of the coordinate ring correspond to affine subvarieties.
Affine subvarieties are precisely those whose coordinate ring is an integral domain.
As maximal ideals are radical, maximal ideals correspond to minimal algebraic sets (those that contain no proper algebraic subsets), which are points in V. If V is an affine variety with coordinate ring
The product of V and W is defined as the algebraic set V × W = V( f1,..., fN, g1,..., gM) in An+m.
There is a one-to-one correspondence between morphisms of affine varieties over an algebraically closed field k, and homomorphisms of coordinate rings of affine varieties over k going in the opposite direction.
Because of this, along with the fact that there is a one-to-one correspondence between affine varieties over k and their coordinate rings, the category of affine varieties over k is dual to the category of coordinate rings of affine varieties over k. The category of coordinate rings of affine varieties over k is precisely the category of finitely-generated, nilpotent-free algebras over k. More precisely, for each morphism φ : V → W of affine varieties, there is a homomorphism φ# : k[W] → k[V] between the coordinate rings (going in the opposite direction), and for each such homomorphism, there is a morphism of the varieties associated to the coordinate rings.
This can be shown explicitly: let V ⊆ kn and W ⊆ km be affine varieties with coordinate rings k[V] = k[X1, ..., Xn] / I and k[W] = k[Y1, ..., Ym] / J respectively.
Hence, each homomorphism φ# : k[W] → k[V] corresponds uniquely to a choice of image for each Yi.
Similarly, for each homomorphism of the coordinate rings, a morphism of the affine varieties can be constructed in the opposite direction.
Mirroring the paragraph above, a homomorphism φ# : k[W] → k[V] sends Yi to a polynomial
Equipped with the structure sheaf described below, an affine variety is a locally ringed space.
Given an affine variety X with coordinate ring A, the sheaf of k-algebras
The key fact, which relies on Hilbert nullstellensatz in the essential way, is the following: Claim —
is a sheaf; indeed, it says if a function is regular (pointwise) on D(f), then it must be in the coordinate ring of D(f); that is, "regular-ness" can be patched together.
This makes the cohomological study of an affine variety non-existent, in a sharp contrast to the projective case in which cohomology groups of line bundles are of central interest.
An affine variety G over an algebraically closed field k is called an affine algebraic group if it has: Together, these define a group structure on the variety.
Using the multiplicative notation, the associativity, identity and inverse laws can be rewritten as: f(gh) = (fg)h, ge = eg = g and gg−1 = g−1g = e. The most prominent example of an affine algebraic group is GLn(k), the general linear group of degree n. This is the group of linear transformations of the vector space kn; if a basis of kn, is fixed, this is equivalent to the group of n×n invertible matrices with entries in k. It can be shown that any affine algebraic group is isomorphic to a subgroup of GLn(k).
