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905:-modules. Geometrically, this is obviously finite since this is a ramified n-sheeted cover of the affine line which degenerates at the origin. By contrast, the inclusion of
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1373:"Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie"
1331:"Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Troisième partie"
1176:
1416:
509:
1504:
419:
1131:, a morphism of schemes is finite if and only if it is proper and quasi-finite. This had been shown by
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is a finite map (in view of the previous definition, because it is between affine varieties).
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925:.) This restricts our geometric intuition to surjective families with finite fibers.
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869:{\displaystyle k/(x^{n}-t)\cong k\oplus k\cdot x\oplus \cdots \oplus k\cdot x^{n-1}}
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Finite morphisms are closed, hence (because of their stability under base change)
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is finite. This corresponds to the following algebraic statement: if
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1193:
1018:-module. Indeed, the generators can be taken to be the elements
1100:. A related statement is that for a finite surjective morphism
731:{\displaystyle {\text{Spec}}(k/(x^{n}-t))\to {\text{Spec}}(k)}
961:
is any morphism of schemes, then the resulting morphism
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Finite morphisms have finite fibers (that is, they are
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theorem of Cohen-Seidenberg in commutative algebra.
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1283:Grothendieck, EGA IV, Part 4, Corollaire 18.12.4.
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1088:). This follows from the fact that for a field
1292:Grothendieck, EGA IV, Part 3, Théorème 8.11.1.
1151:, which follows from the other assumptions if
98:{\displaystyle k\left\hookrightarrow k\left}
1440:
1419:, vol. 52, New York: Springer-Verlag,
1211:
1199:
921:is not finitely generated as a module over
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1223:
399:{\displaystyle V_{i}={\mbox{Spec}}\;B_{i}}
385:
1162:Finite morphisms are both projective and
1050:are finite, as they are locally given by
174:. This definition can be extended to the
323:
243:has an affine neighbourhood V such that
542:{\displaystyle B_{i}\rightarrow A_{i},}
1497:
1070:corresponding to the closed subscheme.
213:between quasiprojective varieties is
1378:Publications Mathématiques de l'IHÉS
1336:Publications Mathématiques de l'IHÉS
468:{\displaystyle f^{-1}(V_{i})=U_{i}}
57:which induces isomorphic inclusion
13:
14:
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478:is an open affine subscheme Spec
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1177:Glossary of algebraic geometry
1149:locally of finite presentation
929:Properties of finite morphisms
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593:is finite if and only if for
1485:The Stacks Project Authors,
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913:is not finite. (Indeed, the
617:is affine, of the form Spec
7:
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1014:is finitely generated as a
994:is finitely generated as a
738:is a finite morphism since
281:{\displaystyle U=f^{-1}(V)}
10:
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1447:Basic Algebraic Geometry 1
176:quasi-projective varieties
1460:10.1007/978-3-642-37956-7
563:finitely generated module
487:, and the restriction of
1304:Stacks Project, Tag 01WG
1270:Stacks Project, Tag 01WG
1254:Stacks Project, Tag 01WG
1238:Stacks Project, Tag 01WG
1187:
1077:. This follows from the
344:is a finite morphism if
1365:Grothendieck, Alexandre
1323:Grothendieck, Alexandre
1214:, p. 62, Def. 1.2.
1202:, p. 60, Def. 1.1.
953:is finite. That is, if
609:, the inverse image of
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597:open affine subscheme
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469:
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236:{\displaystyle y\in Y}
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167:{\displaystyle k\left}
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941:of a finite morphism
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632:For example, for any
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574:. One also says that
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324:Definition by schemes
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46:{\displaystyle X,Y}
1505:Algebraic geometry
1488:The Stacks Project
1412:Algebraic Geometry
1391:10.1007/bf02732123
1349:10.1007/bf02684343
998:-module, then the
986:are (commutative)
915:Laurent polynomial
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500:, which induces a
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18:algebraic geometry
1469:978-0-387-97716-4
1426:978-0-387-90244-9
1407:Hartshorne, Robin
1048:Closed immersions
898:{\displaystyle k}
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502:ring homomorphism
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1096:-algebra is an
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22:finite morphism
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1182:Finite algebra
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1133:Grothendieck
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1086:quasi-finite
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109:, such that
24:between two
21:
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1027:⊗ 1, where
939:base change
328:A morphism
180:regular map
55:regular map
53:is a dense
1499:Categories
1315:References
1157:Noetherian
350:open cover
1510:Morphisms
1385:: 5–361.
1343:: 5–255.
1122:dimension
909:− 0 into
859:−
848:⋅
833:⊕
830:⋯
827:⊕
821:⋅
806:⊕
791:≅
782:−
703:→
691:−
629:-module.
589:In fact,
524:→
429:−
305:→
299::
262:−
228:∈
198:→
192::
79:↪
1444:(2013).
1409:(1977),
1371:(1967).
1329:(1966).
1171:See also
1079:going up
1062:, where
1044:-module.
1435:0463157
1399:0238860
1357:0217086
1129:Deligne
1066:is the
621:, with
601:= Spec
348:has an
342:schemes
1466:
1433:
1423:
1397:
1355:
1164:affine
1075:proper
957:: Z →
580:finite
552:makes
215:finite
1188:Notes
1068:ideal
1040:as a
917:ring
634:field
595:every
582:over
565:over
1464:ISBN
1421:ISBN
1116:and
982:and
937:Any
707:Spec
648:Spec
381:Spec
20:, a
1456:doi
1387:doi
1345:doi
1155:is
1147:is
1127:By
876:as
613:in
605:in
578:is
491:to
352:by
340:of
140:is
16:In
1501::
1462:.
1454:.
1450:.
1431:MR
1429:,
1415:,
1395:MR
1393:.
1383:32
1381:.
1375:.
1367:;
1353:MR
1351:.
1341:28
1339:.
1333:.
1325:;
1143:→
1139::
1112:,
1108:→
1104::
1054:→
974:→
949:→
945::
639:,
586:.
561:a
413:,
336:→
332::
1472:.
1458::
1401:.
1389::
1359:.
1347::
1308:.
1274:.
1258:.
1242:.
1166:.
1159:.
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1094:k
1090:k
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1060:I
1058:/
1056:A
1052:A
1042:B
1038:A
1033:i
1029:a
1024:i
1020:a
1016:C
1012:C
1008:B
1005:⊗
1003:A
996:B
992:A
988:B
984:C
980:A
976:Z
972:Z
968:Y
965:×
963:X
959:Y
955:g
951:Y
947:X
943:f
923:k
919:k
911:A
907:A
893:]
890:t
887:[
884:k
862:1
856:n
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842:t
839:[
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818:]
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711:(
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682:x
678:(
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664:,
661:t
658:[
655:k
652:(
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623:A
619:A
615:X
611:V
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591:f
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576:X
571:i
567:B
558:i
554:A
537:,
532:i
528:A
519:i
515:B
497:i
493:U
489:f
484:i
480:A
461:i
457:U
453:=
450:)
445:i
441:V
437:(
432:1
425:f
411:i
392:i
388:B
376:=
371:i
367:V
346:Y
338:Y
334:X
330:f
308:V
302:U
296:f
276:)
273:V
270:(
265:1
258:f
254:=
251:U
231:Y
225:y
201:Y
195:X
189:f
161:]
158:Y
155:[
151:k
127:]
124:X
121:[
117:k
92:]
89:X
86:[
82:k
75:]
72:Y
69:[
65:k
41:Y
38:,
35:X
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