Incubator escapee wiki

Affine variety

Article Discussion
(Redirected from Affine varieties)

Template:Short description

File:Cubic with double point.svg
A cubic plane curve given by <math>y^2 = x^2(x+1)</math>

In algebraic geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space.

More formally, an affine algebraic set is the set of the common zeros over an algebraically closed field Template:Mvar of some family of polynomials in the polynomial ring <math>k[x_1, \ldots,x_n].</math> An affine variety is an affine algebraic set which is not the union of two smaller algebraic sets; algebraically, this means that (the radical of) the ideal generated by the defining polynomials is prime. One-dimensional affine varieties are called affine algebraic curves, while two-dimensional ones are affine algebraic surfaces.

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 Template:Mvar in which the coefficients are considered, from the algebraically closed field Template:Mvar (containing Template:Mvar) over which the common zeros are considered (that is, the points of the affine algebraic set are in Template:Math). In this case, the variety is said defined over Template:Mvar, and the points of the variety that belong to Template:Math are said Template:Mvar-rational or rational over Template:Mvar. In the common case where Template:Mvar is the field of real numbers, a Template:Mvar-rational point is called a real point.<ref name="ReidUAG">Template:Harvp</ref> When the field Template:Mvar 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 Template:Math has no rational points for any integer Template:Mvar greater than two.

IntroductionEdit

An affine algebraic set is the set of solutions in an algebraically closed field Template:Math of a system of polynomial equations with coefficients in Template:Math. More precisely, if <math>f_1, \ldots, f_m</math> are polynomials with coefficients in Template:Math, they define an affine algebraic set

<math> V(f_1,\ldots, f_m) = \left\{(a_1,\ldots,a_n)\in k^n \;|\;f_1(a_1,\ldots, a_n)=\ldots=f_m(a_1,\ldots, a_n)=0\right\}.</math>

An affine (algebraic) variety is an affine algebraic set that is not the union of two proper affine algebraic subsets. Such an affine algebraic set is often said to be irreducible.

If Template:Math is an affine algebraic set, and Template:Math is the ideal of all polynomials that are zero on Template:Mvar, then the quotient ring <math>R=k[x_1, \ldots, x_n]/I</math> is called the Template:Vanchor of X. If X is an affine variety, then I is prime, so the coordinate ring is an integral domain. The elements of the coordinate ring R are also called the regular functions or the polynomial functions on the variety. They form the ring of regular functions on the variety, or, simply, the ring of the variety; in more technical terms (see Template:Section link), it is the space of global sections of the structure sheaf of X.

The dimension of a variety is an integer associated to every variety, and even to every algebraic set, whose importance relies on the large number of its equivalent definitions (see Dimension of an algebraic variety).

ExamplesEdit

Rational pointsEdit

File:Doubling oriented.svg
A drawing of the real points of the curve Template:Math

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

For an affine variety <math>V\subseteq K^n</math> over an algebraically closed field Template:Math, and a subfield Template:Math of Template:Math, a Template:Math-rational point of Template:Math is a point <math>p\in V\cap k^n.</math> That is, a point of Template:Math whose coordinates are elements of Template:Math. The collection of Template:Math-rational points of an affine variety Template:Math is often denoted <math>V(k).</math> Often, if the base field is the complex numbers Template:Math, points that are Template:Math-rational (where Template:Math is the real numbers) are called real points of the variety, and Template:Math-rational points (Template:Math the rational numbers) are often simply called rational points.

For instance, Template:Math is a Template:Math-rational and an Template:Math-rational point of the variety <math>V = V(x^2+y^2-1)\subseteq\mathbf{C}^2,</math> as it is in Template:Math and all its coordinates are integers. The point Template:Math is a real point of Template:Mvar that is not Template:Math-rational, and <math>(i,\sqrt{2})</math> is a point of Template:Math that is not Template:Math-rational. This variety is called a circle, because the set of its Template:Math-rational points is the unit circle. It has infinitely many Template:Math-rational points that are the points

<math>\left(\frac{1-t^2}{1+t^2},\frac{2t}{1+t^2}\right)</math>

where Template:Mvar is a rational number.

The circle <math>V(x^2+y^2-3)\subseteq\mathbf{C}^2</math> is an example of an algebraic curve of degree two that has no Template:Math-rational point. This can be deduced from the fact that, modulo Template:Math, the sum of two squares cannot be Template:Math.

It can be proved that an algebraic curve of degree two with a Template:Math-rational point has infinitely many other Template:Math-rational points; each such point is the second intersection point of the curve and a line with a rational slope passing through the rational point.

The complex variety <math>V(x^2+y^2+1)\subseteq\mathbf{C}^2</math> has no Template:Math-rational points, but has many complex points.

If Template:Math is an affine variety in Template:Math defined over the complex numbers Template:Math, the Template:Math-rational points of Template:Math can be drawn on a piece of paper or by graphing software. The figure on the right shows the Template:Math-rational points of <math>V(y^2-x^3+x^2+16x)\subseteq\mathbf{C}^2.</math>

Singular points and tangent spaceEdit

Let Template:Mvar be an affine variety defined by the polynomials <math>f_1, \dots, f_r\in k[x_1, \dots, x_n],</math> and <math>a=(a_1, \dots,a_n)</math> be a point of Template:Mvar.

The Jacobian matrix Template:Math of Template:Mvar at Template:Mvar is the matrix of the partial derivatives

<math> \frac{\partial f_j} {\partial {x_i}}(a_1, \dots, a_n).</math>

The point Template:Mvar is regular if the rank of Template:Math equals the codimension of Template:Mvar, and singular otherwise.

If Template:Mvar is regular, the tangent space to Template:Mvar at Template:Mvar is the affine subspace of <math>k^n</math> defined by the linear equations<ref>Template:Harvp</ref>

<math>\sum_{i=1}^n \frac{\partial f_j} {\partial {x_i}}(a_1, \dots, a_n) (x_i - a_i) = 0, \quad j = 1, \dots, r.</math>

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.<ref>Template:Harvp.</ref> A more intrinsic definition which does not use coordinates is given by Zariski tangent space.

The Zariski topologyEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The affine algebraic sets of kn form the closed sets of a topology on kn, called the Zariski topology. This follows from the fact that <math>V(0)=k^n,</math> <math>V(1)=\emptyset,</math> <math>V(S)\cup V(T)=V(ST),</math> and <math>V(S)\cap V(T)=V(S,T)</math> (in fact, a countable intersection of affine algebraic sets is an affine algebraic set).

The Zariski topology can also be described by way of basic open sets, where Zariski-open sets are countable unions of sets of the form <math>U_f = \{p\in k^n:f(p)\neq 0\}</math> for <math>f\in k[x_1,\ldots, x_n].</math> These basic open sets are the complements in kn of the closed sets <math>V(f)=D_f=\{p\in k^n:f(p)=0\},</math> zero loci of a single polynomial. 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.

If V is an affine subvariety of kn the Zariski topology on V is simply the subspace topology inherited from the Zariski topology on kn.

Geometry–algebra correspondenceEdit

The geometric structure of an affine variety is linked in a deep way to the algebraic structure of its coordinate ring. Let I and J be ideals of k[V], the coordinate ring of an affine variety V. Let I(V) be the set of all polynomials in <math>k[x_1, \ldots, x_n],</math> that vanish on V, and let <math>\sqrt{I}</math> 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 <math>k[x_1, \ldots, x_n],</math> where k is an algebraically closed field, <math>I(V(J))=\sqrt{J}.</math>

Radical ideals (ideals that are their own radical) of k[V] correspond to algebraic subsets of V. Indeed, for radical ideals I and J, <math>I\subseteq J</math> if and only if <math>V(J)\subseteq V(I).</math> Hence V(I)=V(J) if and only if I=J. Furthermore, the function taking an affine algebraic set W and returning I(W), the set of all functions that also vanish on all points of W, is the inverse of the function assigning an algebraic set to a radical ideal, by the nullstellensatz. Hence the correspondence between affine algebraic sets and radical ideals is a bijection. The coordinate ring of an affine algebraic set is reduced (nilpotent-free), as an ideal I in a ring R is radical if and only if the quotient ring R/I is reduced.

Prime ideals of the coordinate ring correspond to affine subvarieties. An affine algebraic set V(I) can be written as the union of two other algebraic sets if and only if I=JK for proper ideals J and K not equal to I (in which case <math>V(I)=V(J)\cup V(K)</math>). This is the case if and only if I is not prime. Affine subvarieties are precisely those whose coordinate ring is an integral domain. This is because an ideal is prime if and only if the quotient of the ring by the ideal is an integral domain.

Maximal ideals of k[V] correspond to points of V. If I and J are radical ideals, then <math>V(J)\subseteq V(I)</math> if and only if <math>I\subseteq J.</math> 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 <math>R=k[x_1, \ldots, x_n]/\langle f_1, \ldots, f_m\rangle,</math> this correspondence becomes explicit through the map <math>(a_1,\ldots, a_n) \mapsto \langle \overline{x_1-a_1}, \ldots, \overline{x_n-a_n}\rangle,</math> where <math>\overline{x_i-a_i}</math> denotes the image in the quotient algebra R of the polynomial <math>x_i-a_i.</math> An algebraic subset is a point if and only if the coordinate ring of the subset is a field, as the quotient of a ring by a maximal ideal is a field.

The following table summarizes this correspondence, for algebraic subsets of an affine variety and ideals of the corresponding coordinate ring:

Type of algebraic set Type of ideal Type of coordinate ring
affine algebraic subset radical ideal reduced ring
affine subvariety prime ideal integral domain
point maximal ideal field

Products of affine varietiesEdit

A product of affine varieties can be defined using the isomorphism Template:Math then embedding the product in this new affine space. Let Template:Math and Template:Math have coordinate rings Template:Math and Template:Math respectively, so that their product Template:Math has coordinate ring Template:Math. Let Template:Math be an algebraic subset of Template:Math and Template:Math an algebraic subset of Template:Math Then each Template:Math is a polynomial in Template:Math, and each Template:Math is in Template:Math. The product of Template:Mvar and Template:Mvar is defined as the algebraic set Template:Math in Template:Math The product is irreducible if each Template:Mvar, Template:Mvar is irreducible.<ref>This is because, over an algebraically closed field, the tensor product of integral domains is an integral domain; see integral domain#Properties.</ref>

The Zariski topology on Template:Math 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 Template:Math and Template:Math Hence, polynomials that are in Template:Math but cannot be obtained as a product of a polynomial in Template:Math with a polynomial in Template:Math will define algebraic sets that are closed in the Zariski topology on Template:Math but not in the product topology.

Morphisms of affine varietiesEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

A morphism, or regular map, of affine varieties is a function between affine varieties that is polynomial in each coordinate: more precisely, for affine varieties Template:Math and Template:Math, a morphism from Template:Math to Template:Math is a map Template:Math of the form Template:Math where Template:Math for each Template:Math These are the morphisms in the category of affine varieties.

There is a one-to-one correspondence between morphisms of affine varieties over an algebraically closed field Template:Math and homomorphisms of coordinate rings of affine varieties over Template:Math going in the opposite direction. Because of this, along with the fact that there is a one-to-one correspondence between affine varieties over Template:Math and their coordinate rings, the category of affine varieties over Template:Math is dual to the category of coordinate rings of affine varieties over Template:Math The category of coordinate rings of affine varieties over Template:Math is precisely the category of finitely-generated, nilpotent-free algebras over Template:Math

More precisely, for each morphism Template:Math of affine varieties, there is a homomorphism Template:Math 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 Template:Math and Template:Math be affine varieties with coordinate rings Template:Math and Template:Math respectively. Let Template:Math be a morphism. Indeed, a homomorphism between polynomial rings Template:Math factors uniquely through the ring Template:Math and a homomorphism Template:Math is determined uniquely by the images of Template:Math Hence, each homomorphism Template:Math corresponds uniquely to a choice of image for each Template:Math. Then given any morphism Template:Math from Template:Math to Template:Math a homomorphism can be constructed Template:Math that sends Template:Math to <math>\overline{f_i},</math> where <math>\overline{f_i}</math> is the equivalence class of Template:Math in Template:Math

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 Template:Math sends Template:Math to a polynomial <math>f_i(X_1,\dots,X_n)</math> in Template:Math. This corresponds to the morphism of varieties Template:Math defined by Template:Math

Structure sheafEdit

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 <math>\mathcal{O}_X</math> is defined by letting <math>\mathcal{O}_X(U) = \Gamma(U, \mathcal{O}_X)</math> be the ring of regular functions on U.

Let D(f) = { x | f(x) ≠ 0 } for each f in A. They form a base for the topology of X and so <math>\mathcal{O}_X</math> is determined by its values on the open sets D(f). (See also: sheaf of modules#Sheaf associated to a module.)

The key fact, which relies on Hilbert nullstellensatz in the essential way, is the following: Template:Math theorem Proof:<ref>Template:Harvnb</ref> The inclusion ⊃ is clear. For the opposite, let g be in the left-hand side and <math>J = \{ h \in A | hg \in A \}</math>, which is an ideal. If x is in D(f), then, since g is regular near x, there is some open affine neighborhood D(h) of x such that <math>g \in k[D(h)] = A[h^{-1}]</math>; that is, hm g is in A and thus x is not in V(J). In other words, <math>V(J) \subset \{ x | f(x) = 0 \}</math> and thus the Hilbert nullstellensatz implies f is in the radical of J; i.e., <math>f^n g \in A</math>. <math>\square</math>

The claim, first of all, implies that X is a "locally ringed" space since

<math>\mathcal{O}_{X, x} = \varinjlim_{f(x) \ne 0} A[f^{-1}] = A_{\mathfrak{m}_x}</math>

where <math>\mathfrak{m}_x = \{ f \in A | f(x) = 0 \}</math>. Secondly, the claim implies that <math>\mathcal{O}_X</math> 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.

Hence, <math>(X, \mathcal{O}_X)</math> is a locally ringed space.

Serre's theorem on affinenessEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} A theorem of Serre gives a cohomological characterization of an affine variety; it says an algebraic variety is affine if and only if <math>H^i(X, F) = 0</math> for any <math>i > 0</math> and any quasi-coherent sheaf F on X. (cf. Cartan's theorem B.) 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.

Affine algebraic groupsEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} An affine variety Template:Math over an algebraically closed field Template:Math 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: Template:Math can be written as Template:Math, Template:Math or Template:Math; the inverse Template:Math can be written as Template:Math or Template:Math Using the multiplicative notation, the associativity, identity and inverse laws can be rewritten as: Template:Math, Template:Math and Template:Math.

The most prominent example of an affine algebraic group is Template:Math the general linear group of degree Template:Math This is the group of linear transformations of the vector space Template:Math if a basis of Template:Math is fixed, this is equivalent to the group of Template:Math invertible matrices with entries in Template:Math It can be shown that any affine algebraic group is isomorphic to a subgroup of Template:Math. 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 Template:Math-rational points of an affine algebraic group, where Template:Math is a finite field.

GeneralizationsEdit

  • 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.
  • An open subset of an affine variety is called a quasi-affine variety, so every affine variety is quasi-affine. Any quasi-affine variety is in turn a quasi-projective variety.
  • Affine varieties play the role of local charts for algebraic varieties; that is to say, general algebraic varieties such as projective varieties are obtained by gluing affine varieties. Linear structures that are attached to varieties are also (trivially) affine varieties; e.g., tangent spaces, fibers of algebraic vector bundles.
  • The construction given in Template:Section link allows for a generalization that is used in scheme theory, the modern approach to algebraic geometry. An affine variety is (up to an equivalence of categories) a special case of an affine scheme, a locally-ringed space that is isomorphic to the spectrum of a commutative ring. Each affine variety has an affine scheme associated to it: if Template:Math is an affine variety in Template:Math with coordinate ring Template:Math then the scheme corresponding to Template:Math is Template:Math the set of prime ideals of Template:Math The affine scheme has "classical points", which correspond with points of the variety (and hence maximal ideals of the coordinate ring of the variety), and also a point for each closed subvariety of the variety (these points correspond to prime, non-maximal ideals of the coordinate ring). This creates a more well-defined notion of the "generic point" of an affine variety, by assigning to each closed subvariety an open point that is dense in the subvariety. More generally, an affine scheme is an affine variety if it is reduced, irreducible, and of finite type over an algebraically closed field Template:Math

NotesEdit

Template:Reflist

See alsoEdit

ReferencesEdit

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

|CitationClass=web }}

Retrieved from "https://zalansite.site/mediawiki/index.php?title=Affine_variety&oldid=205690"