Producto de fibra de esquemas

En matemáticas , específicamente en geometría algebraica , el producto fibrado de esquemas es una construcción fundamental. Tiene muchas interpretaciones y casos especiales. Por ejemplo, el producto fibrado describe cómo una variedad algebraica sobre un cuerpo determina una variedad sobre un cuerpo mayor, o la imagen inversa de una familia de variedades, o una fibra de una familia de variedades. El cambio de base es una noción estrechamente relacionada.

La categoría de esquemas constituye un marco amplio para la geometría algebraica. Una filosofía fructífera (conocida como el punto de vista relativo de Grothendieck ) sostiene que gran parte de la geometría algebraica debería desarrollarse para un morfismo de esquemas X → Y (denominado esquema X sobre Y ), en lugar de para un único esquema X. Por ejemplo, en vez de estudiar simplemente curvas algebraicas , se pueden estudiar familias de curvas sobre cualquier esquema base Y. De hecho, ambos enfoques se enriquecen mutuamente.

En particular, un esquema sobre un anillo conmutativo R significa un esquema X junto con un morfismo X → Spec ( R ). La noción antigua de variedad algebraica sobre un cuerpo k es equivalente a un esquema sobre k con ciertas propiedades. (Existen diferentes convenciones sobre qué esquemas deben denominarse "variedades". Una opción estándar es que una variedad sobre un cuerpo k significa un esquema integral separado de tipo finito sobre k . ^{[ 1 ]} )

En general, un morfismo de esquemas X → Y puede imaginarse como una familia de esquemas parametrizados por los puntos de Y. Dado un morfismo de algún otro esquema Z a Y , debería existir una familia de esquemas "de retroceso" sobre Z. Esto es precisamente el producto fibrado X × _Y Z → Z.

Formally: it is a useful property of the category of schemes that the fiber product always exists.^[2] That is, for any morphisms of schemes X → Y and Z → Y, there is a scheme X ×_YZ with morphisms to X and Z, making the diagram

commutative, and which is universal with that property. That is, for any scheme W with morphisms to X and Z whose compositions to Y are equal, there is a unique morphism from W to X ×_YZ that makes the diagram commute. As always with universal properties, this condition determines the scheme X ×_YZ up to a unique isomorphism, if it exists. The proof that fiber products of schemes always do exist reduces the problem to the tensor product of commutative rings (cf. gluing schemes). In particular, when X, Y, and Z are all affine schemes, so X = Spec(A), Y = Spec(B), and Z = Spec(C) for some commutative rings A,B,C, the fiber product is the affine scheme

${\displaystyle X\times _{Y}Z=\operatorname {Spec} (A\otimes _{B}C).}$

The morphism X ×_YZ → Z is called the base change or pullback of the morphism X → Y via the morphism Z → Y.

In some cases, the fiber product of schemes has a right adjoint, the restriction of scalars.

• In the category of schemes over a field k, the productX × Y means the fiber product X ×_kY (which is shorthand for the fiber product over Spec(k)). For example, the product of affine spaces A^m and Aⁿ over a field k is the affine space A^m+n over k.
• For a scheme X over a field k and any field extensionE of k, the base changeX_E means the fiber product X ×_Spec(k) Spec(E). Here X_E is a scheme over E. For example, if X is the curve in the projective planeP²_R over the real numbersR defined by the equation xy² = 7z³, then X_C is the complex curve in P²_C defined by the same equation. Many properties of an algebraic variety over a field k can be defined in terms of its base change to the algebraic closure of k, which makes the situation simpler.
• Let f: X → Y be a morphism of schemes, and let y be a point in Y. Then there is a morphism Spec(k(y)) → Y with image y, where k(y) is the residue field of y. The fiber of f over y is defined as the fiber product X ×_Y Spec(k(y)); this is a scheme over the field k(y).^[3] This concept helps to justify the rough idea of a morphism of schemes X → Y as a family of schemes parametrized by Y.
• Let X, Y, and Z be schemes over a field k, with morphisms X → Y and Z → Y over k. Then the set of k-rational points of the fiber product X ×_YZ is easy to describe:

${\displaystyle (X\times _{Y}Z)(k)=X(k)\times _{Y(k)}Z(k).}$

That is, a k-point of X ×_YZ can be identified with a pair of k-points of X and Z that have the same image in Y. This is immediate from the universal property of the fiber product of schemes.

• If X and Z are closed subschemes of a scheme Y, then the fiber product X ×_YZ is exactly the intersectionX ∩ Z, with its natural scheme structure.^[4] The same goes for open subschemes.

Some important properties P of morphisms of schemes are preserved under arbitrary base change. That is, if X → Y has property P and Z → Y is any morphism of schemes, then the base change X x_YZ → Z has property P. For example, flat morphisms, smooth morphisms, proper morphisms, and many other classes of morphisms are preserved under arbitrary base change.^[5]

The word descent refers to the reverse question: if the pulled-back morphism X x_YZ → Z has some property P, must the original morphism X → Y have property P? Clearly this is impossible in general: for example, Z might be the empty scheme, in which case the pulled-back morphism loses all information about the original morphism. But if the morphism Z → Y is flat and surjective (also called faithfully flat) and quasi-compact, then many properties do descend from Z to Y. Properties that descend include flatness, smoothness, properness, and many other classes of morphisms.^[6] These results form part of Grothendieck's theory of faithfully flat descent.

Example: for any field extension k ⊂ E, the morphism Spec(E) → Spec(k) is faithfully flat and quasi-compact. So the descent results mentioned imply that a scheme X over k is smooth over k if and only if the base change X_E is smooth over E. The same goes for properness and many other properties.

• Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas". Publications Mathématiques de l'IHÉS. 4. doi:10.1007/bf02684778. MR 0217083.
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157