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with certain properties. (There are different conventions for exactly which schemes should be called "varieties". One standard choice is that a variety over a field
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1113:
might be the empty scheme, in which case the pulled-back morphism loses all information about the original morphism. But if the morphism
1137:. Properties that descend include flatness, smoothness, properness, and many other classes of morphisms. These results form part of
60:
845:
303:
up to a unique isomorphism, if it exists. The proof that fiber products of schemes always do exist reduces the problem to the
31:
is a fundamental construction. It has many interpretations and special cases. For example, the fiber product describes how an
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39:
determines a variety over a bigger field, or the pullback of a family of varieties, or a fiber of a family of varieties.
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that makes the diagram commute. As always with universal properties, this condition determines the scheme
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1166:) is faithfully flat and quasi-compact. So the descent results mentioned imply that a scheme
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defined by the same equation. Many properties of an algebraic variety over a field
601:
481:
100:
1358:
1322:
1300:
1077:, and many other classes of morphisms are preserved under arbitrary base change.
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89:
976:. This is immediate from the universal property of the fiber product of schemes.
820:
642:
308:
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is a broad setting for algebraic geometry. A fruitful philosophy (known as
764:). This concept helps to justify the rough idea of a morphism of schemes
618:
20:
1318:
1222:
Grothendieck, EGA I, Théorème 3.2.6; Hartshorne (1977), Theorem II.3.3.
1019:, with its natural scheme structure. The same goes for open subschemes.
193:
Formally: it is a useful property of the category of schemes that the
1109:
have property P? Clearly this is impossible in general: for example,
480:
In some cases, the fiber product of schemes has a right adjoint, the
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can be imagined as a family of schemes parametrized by the points of
426:{\displaystyle X\times _{Y}Z=\operatorname {Spec} (A\otimes _{B}C).}
1377:
525:)). For example, the product of affine spaces A and A over a field
241:
63:) is that much of algebraic geometry should be developed for a
1301:"Éléments de géométrie algébrique: I. Le langage des schémas"
1084:
refers to the reverse question: if the pulled-back morphism
1191:. The same goes for properness and many other properties.
932:{\displaystyle (X\times _{Y}Z)(k)=X(k)\times _{Y(k)}Z(k).}
122:). The older notion of an algebraic variety over a field
1028:
Some important properties P of morphisms of schemes are
92:, one can study families of curves over any base scheme
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169:, there should be a "pullback" family of schemes over
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848:
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always exists. That is, for any morphisms of schemes
521:(which is shorthand for the fiber product over Spec(
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931:
663:can be defined in terms of its base change to the
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1048:is any morphism of schemes, then the base change
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1101:has some property P, must the original morphism
96:. Indeed, the two approaches enrich each other.
257:with that property. That is, for any scheme
1347:, vol. 52, New York: Springer-Verlag,
273:are equal, there is a unique morphism from
88:. For example, rather than simply studying
1333:
1023:
161:. Given a morphism from some other scheme
363:, the fiber product is the affine scheme
493:In the category of schemes over a field
1129:, then many properties do descend from
772:as a family of schemes parametrized by
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1266:
1250:
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1030:preserved under arbitrary base change
61:Grothendieck's relative point of view
1306:Publications Mathématiques de l'IHÉS
1121:is flat and surjective (also called
756:)); this is a scheme over the field
671:, which makes the situation simpler.
173:. This is exactly the fiber product
305:tensor product of commutative rings
84:), rather than for a single scheme
13:
989:are closed subschemes of a scheme
686:be a morphism of schemes, and let
149:In general, a morphism of schemes
16:Construction in algebraic geometry
14:
1407:
1369:
1150:Example: for any field extension
960:can be identified with a pair of
488:Interpretations and special cases
1231:Hartshorne (1977), section II.3.
738:is defined as the fiber product
694:. Then there is a morphism Spec(
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1178:if and only if the base change
126:is equivalent to a scheme over
99:In particular, a scheme over a
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43:is a closely related notion.
1:
1345:Graduate Texts in Mathematics
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1065:has property P. For example,
351:) for some commutative rings
46:
1376:The Stacks Project Authors,
972:that have the same image in
7:
529:is the affine space A over
10:
1412:
993:, then the fiber product
110:together with a morphism
1275:Stacks Project, Tag 02YJ
1259:Stacks Project, Tag 02WE
1243:Stacks Project, Tag 0C4I
1209:Stacks Project, Tag 020D
1194:
791:be schemes over a field
624:defined by the equation
567:means the fiber product
508:means the fiber product
29:fiber product of schemes
1293:Grothendieck, Alexandre
1144:faithfully flat descent
1024:Base change and descent
311:). In particular, when
933:
482:restriction of scalars
427:
269:whose compositions to
934:
823:of the fiber product
428:
234:, making the diagram
1158:, the morphism Spec(
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836:is easy to describe:
600:is the curve in the
370:
213:, there is a scheme
1040:has property P and
65:morphism of schemes
1379:The Stacks Project
1340:Algebraic Geometry
1319:10.1007/bf02684778
929:
815:. Then the set of
596:. For example, if
423:
261:with morphisms to
226:with morphisms to
136:integral separated
25:algebraic geometry
23:, specifically in
1354:978-0-387-90244-9
1335:Hartshorne, Robin
795:, with morphisms
665:algebraic closure
592:is a scheme over
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74:(called a scheme
33:algebraic variety
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1009:intersection
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964:-points of
943:That is, a
706:with image
557:base change
455:base change
251:commutative
140:finite type
41:Base change
21:mathematics
1286:References
947:-point of
138:scheme of
47:Definition
1162:) → Spec(
1080:The word
897:×
857:×
718:) is the
645:curve in
617:over the
406:⊗
396:
378:×
255:universal
134:means an
35:over one
1390:Category
1337:(1977),
1299:(1960).
710:, where
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544:and any
459:pullback
323:are all
53:category
1363:0463157
1327:0217083
1082:descent
643:complex
641:is the
632:, then
499:product
347:= Spec(
343:), and
339:= Spec(
331:= Spec(
57:schemes
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787:, and
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555:, the
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319:, and
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1195:Notes
1170:over
811:over
748:Spec(
734:over
728:fiber
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579:Spec(
573:Spec(
327:, so
307:(cf.
142:over
37:field
1349:ISBN
985:and
968:and
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779:Let
674:Let
393:Spec
265:and
230:and
205:and
116:Spec
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