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A locally
Noetherian scheme is a locally Noetherian formal scheme in the canonical way: the formal completion along itself. In other words, the category of locally Noetherian formal schemes contains all locally Noetherian schemes.
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case. While there have been several definitions of non-Noetherian formal schemes, these encounter technical problems. Consequently, we will only define locally noetherian formal schemes.
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862:{\displaystyle f^{\#}:\Gamma (U,{\mathcal {O}}_{\mathfrak {Y}})\to \Gamma (f^{-1}(U),{\mathcal {O}}_{\mathfrak {X}})}
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admits an open neighborhood isomorphic (as topologically ringed spaces) to the formal spectrum of a noetherian ring.
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of locally noetherian formal schemes is a morphism of them as locally ringed spaces such that the induced map
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have the same underlying topological space but a different structure sheaf. The structure sheaf of Spf
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1184:{\displaystyle {\text{lim}}_{n}{\mathcal {O}}_{{\text{Spec}}A/I^{n}}=\lim _{n}{\widetilde {A/I^{n}}}}
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is a type of space which includes data about its surroundings. Unlike an ordinary
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whose sheaf of rings is a sheaf of topological rings) such that each point of
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The formal completion of a closed subscheme. Consider the closed subscheme
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568:{\displaystyle {\mathcal {O}}_{{\text{Spf}}A}(D_{f})={\widehat {A_{f}}}}
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such that the operations of addition and multiplication are continuous.
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Formal schemes were motivated by and generalize
Zariski's theory of
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1007:{\displaystyle f^{*}({\mathcal {I}}){\mathcal {O}}_{\mathfrak {X}}}
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43:, which is used to deduce theorems of interest for usual schemes.
672:{\displaystyle ({\mathfrak {X}},{\mathcal {O}}_{\mathfrak {X}})}
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is adic, then this property holds for any ideal of definition.
39:. But the concept is also used to prove a theorem such as the
1507:. Contemporary Mathematics. Vol. 314. pp. 187–198.
1441:"Éléments de géométrie algébrique: I. Le langage des schémas"
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Algebraic geometry based on formal schemes is called
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McQuillan, Michael (2002). "Formal formal schemes".
748:{\displaystyle f:{\mathfrak {X}}\to {\mathfrak {Y}}}
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1474:Yasuda, T. (2009). "Non-adic Formal Schemes".
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183:{\displaystyle {\mathcal {J}}^{n}\subseteq V}
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1504:Topology and Geometry: Commemorating SISTAG
1070:. This is preadmissible, and admissible if
384:{\displaystyle A/{\mathcal {J}}_{\lambda }}
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604:{\displaystyle {\widehat {A_{f}}}}
261:. The set of open prime ideals of
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1078:-adically complete. In this case
190:. A linearly topologized ring is
1312:-adic completion. In this case,
711:Morphisms between formal schemes
630:is a topologically ringed space
628:locally noetherian formal scheme
290:{\displaystyle A/{\mathcal {J}}}
76:All rings will be assumed to be
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925:{\displaystyle {\mathfrak {Y}}}
901:{\displaystyle {\mathfrak {X}}}
700:{\displaystyle {\mathfrak {X}}}
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230:{\displaystyle {\mathcal {J}}}
138:{\displaystyle {\mathcal {J}}}
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52:formal holomorphic functions
16:Type of space in mathematics
1410:formal holomorphic function
41:theorem on formal functions
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1395:whose global sections are
1387:. Its global sections are
1304:-adically complete; write
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611:is the completion of the
202:. (In the terminology of
59:formal algebraic geometry
476:It can be shown that if
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932:-adic formal scheme
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1552:Algebraic geometry
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1415:Deformation theory
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1257:. Compare this to
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1437:Dieudonné, Jean
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110:if zero has a
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1557:Scheme theory
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681:ringed space
679:(that is, a
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613:localization
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209:Assume that
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1191:instead of
715:A morphism
626:Finally, a
96:which is a
78:commutative
21:mathematics
1546:Categories
1426:References
958:such that
196:admissible
153:such that
71:Noetherian
65:Definition
1485:0711.0434
1372:~
1215:~
1176:~
1054:and ring
971:∗
821:−
810:Γ
807:→
776:Γ
768:#
736:→
596:^
560:^
457:λ
423:
418:λ
412:←
377:λ
338:λ
175:⊆
80:and with
1439:(1960).
1404:See also
1308:for its
1259:Spec A/I
1084:Spec A/I
1046:Examples
575:, where
480:∈
204:Bourbaki
200:complete
1467:0217083
1314:Spf A=X
1300:is not
1290:I=(y-x)
1241:. Then
501:, then
395:is the
1519:
1465:
1273:about
321:. Let
313:has a
309:. Spf
116:ideals
84:. Let
33:scheme
1480:arXiv
1267:Spf A
1247:Spf A
1243:A/I=k
1239:I=(t)
1080:Spf A
1038:. If
908:is a
118:. An
1517:ISBN
1237:and
1235:A=k]
1116:Spec
881:adic
484:and
436:Spec
112:base
82:unit
27:, a
1509:doi
1490:doi
1455:doi
1397:A/I
1325:lim
1251:(t)
1144:lim
1096:lim
1074:is
1068:a+I
1062:on
883:or
519:Spf
409:lim
301:of
104:is
19:In
1548::
1515:.
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1463:MR
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1298:=k
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1140:=
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1120:A
1109:O
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1072:A
1064:A
1056:A
1052:I
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1024:X
999:X
992:O
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832:U
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813:(
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779:(
773::
764:f
741:Y
731:X
726::
723:f
693:X
667:)
661:X
654:O
648:,
643:X
638:(
620:f
616:A
591:f
587:A
555:f
551:A
544:=
541:)
536:f
532:D
528:(
523:A
512:O
499:f
495:A
490:f
486:D
482:A
478:f
451:J
444:/
440:A
429:O
393:A
371:J
364:/
360:A
332:J
311:A
307:A
303:A
283:J
277:/
273:A
263:A
247:J
223:J
211:A
178:V
170:n
164:J
151:n
147:V
131:J
102:A
94:A
86:A
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