Inmathematics, specifically inalgebraic geometry, thefiber product of schemes is a fundamental construction. It has many interpretations and special cases. For example, the fiber product describes how analgebraic variety over onefield determines a variety over a bigger field, or the pullback of a family of varieties, or a fiber of a family of varieties.Base change is a closely related notion.

Thecategory ofschemes is a broad setting for algebraic geometry. A fruitful philosophy (known asGrothendieck's relative point of view) is that much of algebraic geometry should be developed for amorphism of schemesX →Y (called a schemeXoverY), rather than for a single schemeX. For example, rather than simply studyingalgebraic curves, one can study families of curves over any base schemeY. Indeed, the two approaches enrich each other.

In particular, a scheme over acommutative ringR means a schemeX together with a morphismX →Spec(R). The older notion of an algebraic variety over a fieldk is equivalent to a scheme overk with certain properties. (There are different conventions for exactly which schemes should be called "varieties". One standard choice is that a variety over a fieldk means anintegral separated scheme offinite type overk.^[1])

In general, a morphism of schemesX →Y can be imagined as a family of schemes parametrized by the points ofY. Given a morphism from some other schemeZ toY, there should be a "pullback" family of schemes overZ. This is exactly the fiber productX ×_YZ →Z.

Formally: it is a useful property of the category of schemes that thefiber product always exists.^[2] That is, for any morphisms of schemesX →Y andZ →Y, there is a schemeX ×_YZ with morphisms toX andZ, making the diagram

commutative, and which isuniversal with that property. That is, for any schemeW with morphisms toX andZ whose compositions toY are equal, there is a unique morphism fromW toX ×_YZ that makes the diagram commute. As always with universal properties, this condition determines the schemeX ×_YZ up to a unique isomorphism, if it exists. The proof that fiber products of schemes always do exist reduces the problem to thetensor product of commutative rings (cf.gluing schemes). In particular, whenX,Y, andZ are allaffine schemes, soX = Spec(A),Y = Spec(B), andZ = Spec(C) for some commutative ringsA,B,C, the fiber product is the affine scheme

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

The morphismX ×_YZ →Z is called thebase change orpullback of the morphismX →Y via the morphismZ →Y.

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

• In the category of schemes over a fieldk, theproductX ×Y means the fiber productX ×_kY (which is shorthand for the fiber product over Spec(k)). For example, the product of affine spaces A^m and Aⁿ over a fieldk is the affine space A^m+n overk.
• For a schemeX over a fieldk and anyfield extensionE ofk, thebase changeX_E means the fiber productX ×_Spec(k) Spec(E). HereX_E is a scheme overE. For example, ifX is the curve in theprojective planeP²
  _R over thereal numbersR defined by the equationxy² = 7z³, thenX_C is thecomplex curve inP²
  _C defined by the same equation. Many properties of an algebraic variety over a fieldk can be defined in terms of its base change to thealgebraic closure ofk, which makes the situation simpler.
• Letf:X →Y be a morphism of schemes, and lety be a point inY. Then there is a morphism Spec(k(y)) →Y with imagey, wherek(y) is theresidue field ofy. Thefiber off overy is defined as the fiber productX ×_Y Spec(k(y)); this is a scheme over the fieldk(y).^[3] This concept helps to justify the rough idea of a morphism of schemesX →Y as a family of schemes parametrized byY.
• LetX,Y, andZ be schemes over a fieldk, with morphismsX →Y andZ →Y overk. Then the set ofk-rational points of the fiber productX ×_YZ is easy to describe:

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

That is, ak-point ofX ×_YZ can be identified with a pair ofk-points ofX andZ that have the same image inY. This is immediate from the universal property of the fiber product of schemes.

• IfX andZ are closed subschemes of a schemeY, then the fiber productX ×_YZ is exactly theintersectionX ∩Z, with its natural scheme structure.^[4] The same goes for open subschemes.

Some important properties P of morphisms of schemes arepreserved under arbitrary base change. That is, ifX →Y has property P andZ →Y is any morphism of schemes, then the base changeX 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 worddescent refers to the reverse question: if the pulled-back morphismX x_YZ →Z has some property P, must the original morphismX →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 morphismZ →Y is flat and surjective (also calledfaithfully flat) andquasi-compact, then many properties do descend fromZ toY. Properties that descend include flatness, smoothness, properness, and many other classes of morphisms.^[6] These results form part ofGrothendieck's theory offaithfully flat descent.

Example: for any field extensionk ⊂E, the morphism Spec(E) → Spec(k) is faithfully flat and quasi-compact. So the descent results mentioned imply that a schemeX overk is smooth overk if and only if the base changeX_E is smooth overE. 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

• Scheme theory

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