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is invertible if and only if it is locally free of rank one. Because of this fact, invertible sheaves are closely related to
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39:
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itself. Invertible sheaves are the invertible elements of this monoid. Specifically, if
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276:{\displaystyle L\otimes _{{\mathcal {O}}_{X}}L^{\vee }\to {\mathcal {O}}_{X}}
650:
356:{\displaystyle {\underline {\operatorname {Hom} }}(L,{\mathcal {O}}_{X})}
687:
205:{\displaystyle L\otimes _{{\mathcal {O}}_{X}}M\cong {\mathcal {O}}_{X}}
577:, the study of this functor is a major issue in algebraic geometry.
580:
The direct construction of invertible sheaves by means of data on
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Quite generally, the isomorphism classes of invertible sheaves on
100:-modules form a monoid under the operation of tensor product of
670:"Éléments de géométrie algébrique: I. Le langage des schémas"
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if it satisfies any of the following equivalent conditions:
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Every locally free sheaf of rank one is invertible. If
427:{\displaystyle F\mapsto F\otimes _{{\mathcal {O}}_{X}}L}
91:) be a ringed space. Isomorphism classes of sheaves of
450:, to the point where the two are sometimes conflated.
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511:under tensor product. This group generalises the
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569:. Since it also includes the theory of the
66:, they play a central role in the study of
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473:is the sheaf associated to a rank one
46:which has an inverse with respect to
7:
675:Publications Mathématiques de l'IHÉS
598:Vector bundles in algebraic geometry
551:{\displaystyle \mathrm {Pic} (X)\ }
532:
529:
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25:
62:. Due to their interactions with
442:is a locally ringed space, then
434:is an equivalence of categories.
469:. Then an invertible sheaf on
58:of the topological notion of a
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481:. For example, this includes
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618:Birkhoff-Grothendieck theorem
515:. In general it is written
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649:Stacks Project, tag 01CR,
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54:. It is the equivalent in
303:{\displaystyle L^{\vee }}
283:is an isomorphism, where
215:The natural homomorphism
584:leads to the concept of
662:Grothendieck, Alexandre
487:algebraic number fields
310:denotes the dual sheaf
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113:for this operation is
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147:There exists a sheaf
713:Geometry of divisors
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462:be an affine scheme
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384:-modules defined by
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507:themselves form an
68:algebraic varieties
688:10.1007/bf02684778
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56:algebraic geometry
18:Invertible sheaves
608:First Chern class
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513:ideal class group
483:fractional ideals
475:projective module
366:The functor from
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16:(Redirected from
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571:Jacobian variety
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111:identity element
64:Cartier divisors
36:invertible sheaf
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666:Dieudonné, Jean
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567:Picard functor
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497:Main article:
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50:of sheaves of
48:tensor product
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27:Type of sheaf
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718:Sheaf theory
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613:Picard group
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499:Picard group
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375:-modules to
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44:ringed space
35:
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603:Line bundle
60:line bundle
32:mathematics
707:Categories
624:References
151:such that
141:invertible
139:is called
74:Definition
402:⊗
395:↦
323:_
296:∨
257:→
252:∨
227:⊗
186:≅
163:⊗
668:(1960).
592:See also
454:Examples
696:0217083
52:modules
694:
640:, 5.4.
573:of an
546:
636:EGA 0
561:with
477:over
464:Spec
78:Let (
42:on a
40:sheaf
38:is a
34:, an
565:the
458:Let
684:doi
563:Pic
485:of
320:Hom
70:.
30:In
709::
692:MR
690:.
682:.
678:.
672:.
664:;
588:.
82:,
698:.
686::
680:4
652:.
638:I
582:X
543:)
540:X
537:(
533:c
530:i
527:P
505:X
479:R
471:X
466:R
460:X
444:L
440:X
422:L
415:X
409:O
398:F
392:F
381:X
377:O
372:X
368:O
363:.
351:)
346:X
340:O
334:,
331:L
328:(
292:L
269:X
263:O
248:L
240:X
234:O
223:L
212:.
198:X
192:O
183:M
176:X
170:O
159:L
149:M
137:L
132:X
128:O
124:L
119:X
115:O
106:X
102:O
97:X
93:O
88:X
84:O
80:X
20:)
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