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934:"ĂlĂ©ments de gĂ©omĂ©trie algĂ©brique (rĂ©digĂ©s avec la collaboration de Jean DieudonnĂ©) : IV. Ătude locale des schĂ©mas et des morphismes de schĂ©mas, TroisiĂšme partie"
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902:"ĂlĂ©ments de gĂ©omĂ©trie algĂ©brique (rĂ©digĂ©s avec la collaboration de Jean DieudonnĂ©) : II. Ătude globale Ă©lĂ©mentaire de quelques classes de morphismes"
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SĂ©minaire de GĂ©omĂ©trie AlgĂ©brique du Bois Marie - 1960-61 - RevĂȘtements Ă©tales et groupe fondamental - (SGA 1) (Documents MathĂ©matiques
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locally of finite presentation is finite. Indeed, a morphism is finite if and only if it is proper and locally quasi-finite. Since
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Quasi-finiteness is preserved by base change. The composite and fiber product of quasi-finite morphisms is quasi-finite.
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II 6.2 because it makes it possible to give an algebraic characterization of quasi-finiteness in terms of
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1 and did not include the finite type hypothesis. This hypothesis was added to the definition in
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are of finite type and finite type morphisms are quasi-compact one may omit the qualification
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878:(in French) (Updated ed.). Société Mathématique de France. xviii+327.
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where the first morphism is an open immersion and the second is finite. (
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be quasi-finite, separated and of finite presentation. Then
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and satisfies any of the following equivalent conditions:
201:{\displaystyle {\mathcal {O}}_{X,x}\otimes \kappa (f(x))}
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450:is quasi-finite if any of the following are true:
250:Quasi-finite morphisms were originally defined by
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574:{\displaystyle {\mathcal {O}}_{f^{-1}(f(x)),x}}
365:locally quasi-finite morphism is quasi-finite.
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303:if there exist open affine neighborhoods
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686:{\displaystyle X\hookrightarrow X'\to Y}
357:if it is quasi-finite at every point in
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385:is quasi-finite, then the induced map
16:Type of merphism in algebraic geometry
712:quasi-finite fundamental group scheme
377:, the following properties are true.
7:
938:Publications MathĂ©matiques de l'IHĂS
906:Publications MathĂ©matiques de l'IHĂS
619:are quasi-finite. A quasi-finite
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239:{\displaystyle \kappa (f(x))}
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208:is finitely generated over
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755:EGA II, Proposition 6.2.4.
640:is the following: Suppose
777:EGA II, Corollaire 6.1.7.
648:and quasi-separated. Let
128:)) scheme. (Here κ(
66:is isolated in its fiber
730:EGA II, DĂ©finition 6.2.3
926:Grothendieck, Alexandre
894:Grothendieck, Alexandre
868:Grothendieck, Alexandre
269:For a general morphism
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634:A generalized form of
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252:Alexander Grothendieck
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959:Morphisms of schemes
826:"Definition 29.15.1"
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637:Zariski Main Theorem
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355:locally quasi-finite
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512:is quasi-finite at
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120:is a finite κ(
918:10.1007/bf02699291
853:, ThéorÚme 8.12.6.
830:The Stacks Project
805:The Stacks Project
790:, ThéorÚme 8.11.1.
768:, ThéorÚme 17.4.1.
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508:. Conversely, if
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833:. Retrieved
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801:"Lemma 02LS"
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739:EGA III, Err
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283:and a point
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110:Spec κ(
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58:Every point
50:if it is of
48:quasi-finite
47:
39:
35:
31:
18:
656:factors as
496:at a point
52:finite type
25:mathematics
861:References
810:31 January
494:unramified
369:Properties
912:: 5â222.
835:15 August
678:→
667:↪
605:)), then
538:−
436:, and if
427:g :
216:κ
178:κ
175:⊗
953:Category
944:: 5â255.
932:(1966).
900:(1961).
705:See also
674:′
431:→
392:between
344:→
340: :
278:→
274: :
38:→
34: :
29:morphism
629:locally
500:, then
44:schemes
882:
849:EGA IV
786:EGA IV
764:EGA IV
469:×
264:stalks
106:×
743:, 20.
718:Notes
361:. A
102:)) =
880:ISBN
837:2023
812:2022
710:The
311:and
46:is
27:, a
914:doi
741:III
701:.)
644:is
492:is
488:If
425:If
410:If
399:If
390:red
381:If
353:is
315:of
307:of
299:at
287:in
260:EGA
256:SGA
254:in
143:of
85:of
62:of
42:of
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