Affine variety

In algebraic geometry, an affine algebraic set is the set of the common zeros over an algebraically closed field $k$ of some family of polynomials in the polynomial ring $k[X_{1},\ldots ,X_{n}].$ An affine variety or affine algebraic variety, is an affine algebraic set such that the ideal generated by the defining polynomials is prime.

Some texts use the term variety for any algebraic set, and irreducible variety an algebraic set whose defining ideal is prime (affine variety in the above sense).

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 $K n$ ). In this case, the variety is said defined over $k$ , and the points of the variety that belong to $k n$ are said $k$ -rational or rational over $k$ . In the common case where $k$ is the field of real numbers, a $k$ -rational point is called a real point.^[1] When the field $k$ is not specified, a rational point is a point that is rational over the rational numbers. For example, Fermat's Last Theorem asserts that the affine algebraic variety (it is a curve) defined by $x n + y n - 1 = 0$ has no rational points for any integer $n$ greater than two.

The complement of a hypersurface in an affine variety $X$ (that is $X \ { f = 0 }$ for some polynomial $f$ ) is affine. Its defining equations are obtained by by $f$ the defining ideal of $X$ . The coordinate ring is thus the localization $k[X][f^{-1}]$ .

saturating

In particular, $\mathbb {C} -0$ (the affine line with the origin removed) is affine.

On the other hand, $\mathbb {C} ^{2}-0$ (the affine plane with the origin removed) is not an affine variety; cf. .

Hartogs' extension theorem

The subvarieties of codimension one in the affine space $k^{n}$ are exactly the hypersurfaces, that is the varieties defined by a single polynomial.

The of an irreducible affine variety is affine; the coordinate ring of the normalization is the integral closure of the coordinate ring of the variety. (Similarly, the normalization of a projective variety is a projective variety.)

normalization

Products of affine varieties[edit]

A product of affine varieties can be defined using the isomorphism $A n \times A m = A n + m,$ then embedding the product in this new affine space. Let $A n$ and $A m$ have coordinate rings $k [x 1,..., x n]$ and $k [y 1,..., y m]$ respectively, so that their product $A n + m$ has coordinate ring $k [x 1,..., x n, y 1,..., y m]$ . Let $V = V (f 1,..., f N)$ be an algebraic subset of $A n,$ and $W = V (g 1,..., g M)$ an algebraic subset of $A m .$ Then each $f i$ is a polynomial in $k [x 1,..., x n]$ , and each $g j$ is in $k [y 1,..., y m]$ . The product of $V$ and $W$ is defined as the algebraic set $V \times W = V (f 1,..., f N, g 1,..., g M)$ in $A n + m .$ The product is irreducible if each $V$ , $W$ is irreducible.^[4]

The Zariski topology on $A n \times A m$ is not the topological product of the Zariski topologies on the two spaces. Indeed, the product topology is generated by products of the basic open sets $U f = A n - V (f)$ and $T g = A m - V (g).$ Hence, polynomials that are in $k [x 1,..., x n, y 1,..., y m]$ but cannot be obtained as a product of a polynomial in $k [x 1,..., x n]$ with a polynomial in $k [y 1,..., y m]$ will define algebraic sets that are in the Zariski topology on $A n \times A m,$ but not in the product topology.

A multiplication $μ : G \times G \to G$ , which is a regular morphism that follows the axiom—that is, such that $μ (μ (f, g), h) = μ (f, μ (g, h))$ for all points $f$ , $g$ and $h$ in $G;$

associativity

An identity element $e$ such that $μ (e, g) = μ (g, e) = g$ for every $g$ in $G;$

An inverse morphism, a regular bijection $ι : G \to G$ such that $μ (ι (g), g) = μ (g, ι (g)) = e$ for every $g$ in $G .$

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. The above morphisms are often written using ordinary group notation: $μ (f, g)$ can be written as $f + g$ , $f \cdot g,$ or $fg$ ; the inverse $ι (g)$ can be written as $- g$ or $g -1 .$ Using the multiplicative notation, the associativity, identity and inverse laws can be rewritten as: $f (gh) = (fg) h$ , $ge = eg = g$ and $gg -1 = g -1 g = e$ .

The most prominent example of an affine algebraic group is $GL n (k),$ the general linear group of degree $n .$ This is the group of linear transformations of the vector space $k n;$ if a basis of $k n,$ is fixed, this is equivalent to the group of $n \times n$ invertible matrices with entries in $k .$ It can be shown that any affine algebraic group is isomorphic to a subgroup of $GL n (k)$ . For this reason, affine algebraic groups are often called linear algebraic groups.

Affine algebraic groups play an important role in the classification of finite simple groups, as the groups of Lie type are all sets of $F q$ -rational points of an affine algebraic group, where $F q$ is a finite field.

If an author requires the base field of an affine variety to be algebraically closed (as this article does), then irreducible affine algebraic sets over non-algebraically closed fields are a generalization of affine varieties. This generalization notably includes affine varieties over the .

real numbers

Algebraic variety

Affine scheme

Representations on coordinate rings

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Milne, James S.

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The original article was written as a partial human translation of the corresponding French article.