1430:
971:
is closed. In algebraic geometry, the above formulation is used because a scheme which is a
Hausdorff space is necessarily empty or zero-dimensional. The difference between the topological and algebro-geometric context comes from the topological structure of the fiber product (in the category of
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with itself is a closed immersion. Emphasizing the relative point of view, one might equivalently define a scheme to be separated if the unique morphism
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is separated, because the diagonal corresponds to the surjective map of rings (hence is a closed immersion of schemes):
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60:
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be a scheme obtained by identifying two affine lines through the identity map except at the origins (see
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is by intersecting (restricting) their cartesian product with (to) the diagonal: precisely,
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1103:{\displaystyle A\otimes _{\mathbb {Z} }A\rightarrow A,a\otimes a'\mapsto a\cdot a'}
961:{\displaystyle Y{\stackrel {\Delta }{\longrightarrow }}Y\times Y,\,y\mapsto (y,y)}
1429:
765:
as a closed subscheme — in other words, the diagonal morphism is a
762:
1139:). It is not separated. Indeed, the image of the diagonal morphism
1171:
image has two origins, while its closure contains four origins.
1020:, which is different from the product of topological spaces.
462:
if and only if the diagonal embedding is an open immersion.
874:{\displaystyle X\rightarrow {\textrm {Spec}}(\mathbb {Z} )}
1013:{\displaystyle X\times _{{\textrm {Spec}}(\mathbb {Z} )}X}
105:
is a morphism determined by the universal property of the
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595:{\displaystyle X\times _{k}X=\{(x,y)\in X\times X\}}
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344:. The diagonal embedding is the graph morphism of
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133:
94:
43:
1269:{\displaystyle A\cdot B=\delta ^{*}(A\times B)}
518:{\displaystyle p:X\to \operatorname {Spec} (k)}
1465:
1306:is the pullback along the diagonal embedding
8:
589:
559:
1402:, vol. 52, New York: Springer-Verlag,
1472:
1458:
1377:
640:{\displaystyle \delta :X\to X\times _{k}X}
95:{\displaystyle \delta :X\to X\times _{S}X}
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83:
62:
24:
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1337:{\displaystyle \delta :X\to X\times X}
685:; whence the name diagonal morphism.
525:the structure map. Then, identifying
458:locally of finite presentation is an
7:
1426:
1424:
1444:. You can help Knowledge (XXG) by
917:
290:{\displaystyle X\to X\times _{S}Y}
14:
1428:
422:) if the diagonal morphism is a
16:In algebraic geometry, given a
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1263:
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1164:{\displaystyle S\to S\times S}
1149:
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678:{\displaystyle x\mapsto (x,x)}
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268:
258:, the graph morphism of it is
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172:
73:
35:
1:
1400:Graduate Texts in Mathematics
134:{\displaystyle X\times _{S}X}
1179:A classic way to define the
895:iff the diagonal embedding
181:{\displaystyle 1_{X}:X\to X}
1299:{\displaystyle \delta ^{*}}
772:As a consequence, a scheme
470:As an example, consider an
1512:
1423:
1175:Use in intersection theory
476:algebraically closed field
218:It is a special case of a
1491:Algebraic geometry stubs
451:{\displaystyle p:X\to S}
411:{\displaystyle p:X\to S}
247:{\displaystyle f:X\to Y}
149:applied to the identity
44:{\displaystyle p:X\to S}
1440:–related article is a
1338:
1300:
1270:
1206:
1165:
1137:gluing scheme#Examples
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796:when the diagonal of
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364:{\displaystyle 1_{X}}
339:
337:{\displaystyle 1_{X}}
312:
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208:{\displaystyle 1_{X}}
183:
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97:
46:
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1181:intersection product
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1119:
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902:
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529:with the set of its
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61:
23:
1205:{\displaystyle A,B}
460:unramified morphism
426:. Also, a morphism
222:: given a morphism
18:morphism of schemes
1496:Algebraic geometry
1438:algebraic geometry
1395:Algebraic Geometry
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1296:
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741:with itself along
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707:
695:separated morphism
689:Separated morphism
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533:-rational points,
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448:
420:separated morphism
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131:
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41:
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1409:978-0-387-90244-9
1390:Hartshorne, Robin
1359:Diagonal morphism
1354:regular embedding
1128:{\displaystyle S}
991:
921:
886:topological space
857:
833:{\displaystyle X}
809:{\displaystyle X}
785:{\displaystyle X}
754:{\displaystyle f}
734:{\displaystyle f}
710:{\displaystyle f}
472:algebraic variety
317:and the identity
310:{\displaystyle f}
188:and the identity
53:diagonal morphism
1503:
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1381:
1380:, Example 4.0.1.
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1185:algebraic cycles
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767:closed immersion
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424:closed immersion
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380:separated scheme
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1378:Hartshorne 1977
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374:By definition,
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1214:smooth variety
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884:Notice that a
881:is separated.
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818:scheme product
805:
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717:such that the
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697:is a morphism
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220:graph morphism
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719:fiber product
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107:fiber product
89:
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76:
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67:
64:
57:
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55:
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38:
32:
29:
26:
19:
1446:expanding it
1435:
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1114:
1035:
1028:
1024:
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970:
888:
883:
817:
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771:
766:
694:
692:
647:is given as
530:
526:
478:
469:
419:
383:
379:
375:
373:
255:
217:
146:
142:
104:
52:
15:
816:within the
466:Explanation
297:induced by
1485:Categories
1365:References
1329:×
1323:→
1314:δ
1292:∗
1288:δ
1258:×
1247:∗
1243:δ
1233:⋅
1156:×
1150:→
1090:⋅
1084:↦
1073:⊗
1061:→
1047:⊗
984:×
972:schemes)
941:↦
928:×
918:Δ
913:⟶
893:Hausdorff
851:→
794:separated
658:↦
626:×
619:→
610:δ
584:×
578:∈
545:×
504:
498:→
443:→
403:→
276:×
269:→
239:→
173:→
120:×
81:×
74:→
65:δ
36:→
1392:(1977),
1348:See also
1097:′
1080:′
763:diagonal
761:has its
474:over an
1418:0463157
1027:scheme
1416:
1406:
1279:where
1029:Spec A
1025:affine
51:, the
1436:This
1212:on a
418:is a
382:over
378:is a
254:over
1442:stub
1404:ISBN
1115:Let
1023:Any
990:Spec
856:Spec
602:and
501:Spec
481:and
145:and
1183:of
891:is
820:of
792:is
721:of
141:of
1487::
1414:MR
1412:,
1398:,
1344:.
769:.
693:A
371:.
215:.
1473:e
1466:t
1459:v
1448:.
1332:X
1326:X
1320:X
1317::
1264:)
1261:B
1255:A
1252:(
1239:=
1236:B
1230:A
1217:X
1200:B
1197:,
1194:A
1159:S
1153:S
1147:S
1123:S
1111:.
1094:a
1087:a
1077:a
1070:a
1067:,
1064:A
1058:A
1052:Z
1043:A
1008:X
1003:)
999:Z
995:(
980:X
956:)
953:y
950:,
947:y
944:(
938:y
934:,
931:Y
925:Y
906:Y
889:Y
869:)
865:Z
861:(
848:X
828:X
804:X
780:X
749:f
729:f
705:f
673:)
670:x
667:,
664:x
661:(
655:x
635:X
630:k
622:X
616:X
613::
590:}
587:X
581:X
575:)
572:y
569:,
566:x
563:(
560:{
557:=
554:X
549:k
541:X
531:k
527:X
513:)
510:k
507:(
495:X
492::
489:p
479:k
446:S
440:X
437::
434:p
406:S
400:X
397::
394:p
386:(
384:S
376:X
357:X
353:1
330:X
326:1
305:f
285:Y
280:S
272:X
266:X
256:S
242:Y
236:X
233::
230:f
201:X
197:1
176:X
170:X
167::
162:X
158:1
147:p
143:p
129:X
124:S
116:X
90:X
85:S
77:X
71:X
68::
39:S
33:X
30::
27:p
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