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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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71:
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380:
1124:
might be the empty scheme, in which case the pulled-back morphism loses all information about the original morphism. But if the morphism
1148:. Properties that descend include flatness, smoothness, properness, and many other classes of morphisms. These results form part of
856:
314:
up to a unique isomorphism, if it exists. The proof that fiber products of schemes always do exist reduces the problem to the
42:
is a fundamental construction. It has many interpretations and special cases. For example, the fiber product describes how an
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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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1154:
67:
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75:
47:
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35:
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1177:) 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
17:
1311:
1088:, and many other classes of morphisms are preserved under arbitrary base change.
987:. This is immediate from the universal property of the fiber product of schemes.
629:
31:
1284:
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1252:
1218:
252:
70:
is a broad setting for algebraic geometry. A fruitful philosophy (known as
775:). This concept helps to justify the rough idea of a morphism of schemes
1329:
1233:
Grothendieck, EGA I, Théorème 3.2.6; Hartshorne (1977), Theorem II.3.3.
1030:, with its natural scheme structure. The same goes for open subschemes.
204:
Formally: it is a useful property of the category of schemes that the
1120:
have property P? Clearly this is impossible in general: for example,
491:
In some cases, the fiber product of schemes has a right adjoint, the
168:
can be imagined as a family of schemes parametrized by the points of
437:{\displaystyle X\times _{Y}Z=\operatorname {Spec} (A\otimes _{B}C).}
1388:
536:)). For example, the product of affine spaces A and A over a field
74:) is that much of algebraic geometry should be developed for a
1312:"Éléments de géométrie algébrique: I. Le langage des schémas"
1095:
refers to the reverse question: if the pulled-back morphism
1202:. The same goes for properness and many other properties.
943:{\displaystyle (X\times _{Y}Z)(k)=X(k)\times _{Y(k)}Z(k).}
133:). The older notion of an algebraic variety over a field
1039:
Some important properties P of morphisms of schemes are
103:, one can study families of curves over any base scheme
180:, there should be a "pullback" family of schemes over
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383:
208:
always exists. That is, for any morphisms of schemes
532:(which is shorthand for the fiber product over Spec(
942:
674:can be defined in terms of its base change to the
436:
1059:is any morphism of schemes, then the base change
1112:has some property P, must the original morphism
107:. Indeed, the two approaches enrich each other.
8:
268:with that property. That is, for any scheme
1358:, vol. 52, New York: Springer-Verlag,
284:are equal, there is a unique morphism from
99:. For example, rather than simply studying
172:. Given a morphism from some other scheme
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870:
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419:
391:
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374:, the fiber product is the affine scheme
504:In the category of schemes over a field
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1140:, then many properties do descend from
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1041:preserved under arbitrary base change
72:Grothendieck's relative point of view
7:
1317:Publications Mathématiques de l'IHÉS
1132:is flat and surjective (also called
767:)); this is a scheme over the field
682:, which makes the situation simpler.
184:. This is exactly the fiber product
316:tensor product of commutative rings
95:), rather than for a single scheme
1000:are closed subschemes of a scheme
697:be a morphism of schemes, and let
160:In general, a morphism of schemes
27:Construction in algebraic geometry
25:
1161:Example: for any field extension
971:can be identified with a pair of
499:Interpretations and special cases
1242:Hartshorne (1977), section II.3.
749:is defined as the fiber product
705:. Then there is a morphism Spec(
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1189:if and only if the base change
137:is equivalent to a scheme over
110:In particular, a scheme over a
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903:
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54:is a closely related notion.
1:
1356:Graduate Texts in Mathematics
1076:has property P. For example,
362:) for some commutative rings
1387:The Stacks Project Authors,
983:that have the same image in
540:is the affine space A over
1423:
1004:, then the fiber product
121:together with a morphism
1286:Stacks Project, Tag 02YJ
1270:Stacks Project, Tag 02WE
1254:Stacks Project, Tag 0C4I
1220:Stacks Project, Tag 020D
802:be schemes over a field
635:defined by the equation
578:means the fiber product
519:means the fiber product
40:fiber product of schemes
1304:Grothendieck, Alexandre
1155:faithfully flat descent
1035:Base change and descent
322:). In particular, when
944:
493:restriction of scalars
438:
280:whose compositions to
945:
834:of the fiber product
439:
245:, making the diagram
1169:, the morphism Spec(
857:
847:is easy to describe:
611:is the curve in the
381:
224:, there is a scheme
1051:has property P and
76:morphism of schemes
1390:The Stacks Project
1351:Algebraic Geometry
1330:10.1007/bf02684778
940:
826:. Then the set of
607:. For example, if
434:
272:with morphisms to
237:with morphisms to
147:integral separated
36:algebraic geometry
34:, specifically in
18:Product of schemes
1365:978-0-387-90244-9
1346:Hartshorne, Robin
806:, with morphisms
676:algebraic closure
603:is a scheme over
480:via the morphism
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85:(called a scheme
44:algebraic variety
16:(Redirected from
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1043:. That is, if
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1020:intersection
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630:real numbers
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975:-points of
954:That is, a
717:with image
568:base change
466:base change
262:commutative
151:finite type
52:Base change
32:mathematics
1297:References
958:-point of
149:scheme of
58:Definition
1173:) → Spec(
1091:The word
908:×
868:×
729:) is the
656:curve in
628:over the
417:⊗
407:
389:×
266:universal
145:means an
46:over one
1401:Category
1348:(1977),
1310:(1960).
721:, where
594:). Here
555:and any
470:pullback
334:are all
64:category
1374:0463157
1338:0217083
1093:descent
654:complex
652:is the
643:, then
510:product
358:= Spec(
354:), and
350:= Spec(
342:= Spec(
68:schemes
1372:
1362:
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798:, and
737:. The
566:, the
508:, the
330:, and
38:, the
1206:Notes
1181:over
822:over
759:Spec(
745:over
739:fiber
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590:Spec(
584:Spec(
338:, so
318:(cf.
153:over
48:field
1360:ISBN
996:and
979:and
814:and
790:Let
685:Let
404:Spec
276:and
241:and
216:and
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