271:
in the
Zariski topology. This causes problems when attempting to define left derived functors of a right exact functor (such as Tor). This can sometimes be done by ad hoc means: for example, the left derived functors of Tor can be defined using a flat resolution rather than a projective one, but it
103:
sufficed to found the theory. The other classes of sheaves are historically older notions. The abstract framework for defining cohomology and derived functors does not need them. However, in most concrete situations, resolutions by acyclic sheaves are often easier to construct. Acyclic sheaves
267:, is not used much, because in a general category of sheaves there are not enough of them: not every sheaf is the quotient of a projective sheaf, and in particular projective resolutions do not always exist. This is the case, for example, when looking at the category of sheaves on
357:. Typical examples are the sheaf of germs of continuous real-valued functions over such a space, or smooth functions over a smooth (paracompact Hausdorff) manifold, or modules over these sheaves of rings. Also, fine sheaves over paracompact Hausdorff spaces are soft and acyclic.
255:
For technical purposes, injective sheaves are usually superior to the other classes of sheaves mentioned above: they can do almost anything the other classes can do, and their theory is simpler and more general. In fact, injective sheaves are flabby
497:
871:
623:
244:
The category of abelian sheaves has enough injective objects: this means that any sheaf is a subsheaf of an injective sheaf. This result of
Grothendieck follows from the existence of a
775:
240:
716:
658:
309:
213:
189:
165:
141:
530:
390:
260:), soft, and acyclic. However, there are situations where the other classes of sheaves occur naturally, and this is especially true in concrete computational situations.
272:
takes some work to show that this is independent of the resolution. Not all categories of sheaves run into this problem; for instance, the category of sheaves on an
402:
736:
793:
895:
Flasque sheaves are useful because (by definition) their sections extend. This means that they are some of the simplest sheaves to handle in terms of
346:
we can find a family of homomorphisms from the sheaf to itself with sum 1 such that each homomorphism is 0 outside some element of the open cover.
538:
252:). This is enough to show that right derived functors of any left exact functor exist and are unique up to canonical isomorphism.
993:
966:
900:
908:
360:
One can find a resolution of a sheaf on a smooth manifold by fine sheaves using the
Alexander–Spanier resolution.
318:
The cohomology groups of any sheaf can be calculated from any acyclic resolution of it (this goes by the name of
143:
is a sheaf that is an injective object of the category of abelian sheaves; in other words, homomorphisms from
748:
1074:
899:. Any sheaf has a canonical embedding into the flasque sheaf of all possibly discontinuous sections of the
17:
1069:
218:
697:
639:
290:
194:
170:
146:
122:
513:
373:
105:
1010:
90:
532:
can thus be computed as the cohomology of the complex of globally defined differential forms:
492:{\displaystyle 0\to \mathbb {R} \to C_{X}^{0}\to C_{X}^{1}\to \cdots \to C_{X}^{\dim X}\to 0.}
1079:
1044:
1003:
889:
319:
249:
8:
896:
881:
785:
692:
339:
58:
885:
866:{\displaystyle r_{U\subseteq V}:\Gamma (V,{\mathcal {F}})\to \Gamma (U,{\mathcal {F}})}
721:
393:
1032:
989:
962:
739:
503:
1022:
954:
912:
268:
99:
94:
43:
903:, and by repeating this we can find a canonical flasque resolution for any sheaf.
1040:
999:
925:
47:
981:
958:
1063:
1036:
1027:
273:
39:
953:. Graduate Texts in Mathematics. Vol. 94. pp. 186, 181, 178, 170.
1054:
86:
248:
of the category (it can be written down explicitly, and is related to the
781:
350:
51:
31:
877:
85:. In the history of the subject they were introduced before the 1957 "
364:
1050:
951:
618:{\displaystyle H^{i}(X,\mathbb {R} )=H^{i}(C_{X}^{\bullet }(X)).}
502:
This is a resolution, i.e. an exact complex of sheaves by the
675:
Soft sheaves are acyclic over paracompact
Hausdorff spaces.
315:
is one such that all higher sheaf cohomology groups vanish.
104:
therefore serve for computational purposes, for example the
911:
by means of flasque sheaves, are one approach to defining
370:. There is the following resolution of the constant sheaf
928:
word that has sometimes been translated into
English as
57:
There is a further group of related concepts applied to
42:
are used to construct the resolutions needed to define
1013:(1957), "Sur quelques points d'algèbre homologique",
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342:"; more precisely for any open cover of the space
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159:
135:
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986:Topologie algébrique et théorie des faisceaux
1051:"Sheaf cohomology and injective resolutions"
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678:
1026:
562:
518:
413:
378:
980:
349:Fine sheaves are usually only used over
770:{\displaystyle U\subseteq V\subseteq X}
27:Mathematical object in sheaf cohomology
14:
1062:
948:
918:Flasque sheaves are soft and acyclic.
664:is one such that any section over any
672:can be extended to a global section.
191:can always be extended to any sheaf
111:
363:As an application, consider a real
24:
855:
841:
830:
816:
742:on which the sheaf is defined and
703:
645:
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200:
176:
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25:
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718:with the following property: if
392:by the fine sheaves of (smooth)
1015:The Tohoku Mathematical Journal
628:
325:
235:{\displaystyle {\mathcal {A}}.}
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860:
844:
838:
835:
819:
711:{\displaystyle {\mathcal {F}}}
653:{\displaystyle {\mathcal {F}}}
609:
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566:
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483:
459:
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409:
304:{\displaystyle {\mathcal {F}}}
208:{\displaystyle {\mathcal {B}}}
184:{\displaystyle {\mathcal {F}}}
160:{\displaystyle {\mathcal {A}}}
136:{\displaystyle {\mathcal {F}}}
13:
1:
935:
276:contains enough projectives.
525:{\displaystyle \mathbb {R} }
385:{\displaystyle \mathbb {R} }
7:
10:
1096:
959:10.1007/978-1-4757-1799-0
949:Warner, Frank W. (1983).
679:Flasque or flabby sheaves
93:, which showed that the
1011:Grothendieck, Alexander
106:Leray spectral sequence
1028:10.2748/tmj/1178244839
867:
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712:
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91:Alexander Grothendieck
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506:. The cohomology of
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320:De Rham-Weil theorem
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250:subobject classifier
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1075:Homological algebra
905:Flasque resolutions
897:homological algebra
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340:partitions of unity
1070:Algebraic geometry
988:, Paris: Hermann,
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394:differential forms
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265:projective sheaves
263:The dual concept,
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133:
1017:, Second Series,
995:978-2-7056-1252-8
968:978-1-4419-2820-7
740:topological space
731:{\displaystyle X}
353:Hausdorff spaces
112:Injective sheaves
36:injective sheaves
16:(Redirected from
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913:sheaf cohomology
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280:Acyclic sheaves
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1021:(2): 119–221,
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504:Poincaré lemma
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40:abelian groups
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987:
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979:
978:
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945:
941:
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923:
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764:
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685:flasque sheaf
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338:is one with "
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286:acyclic sheaf
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274:affine scheme
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1080:Sheaf theory
1055:MathOverflow
1018:
1014:
985:
950:
944:
929:
921:
920:
917:
904:
894:
875:
782:open subsets
779:
738:is the base
689:flabby sheaf
688:
684:
682:
674:
669:
665:
661:
634:
632:
629:Soft sheaves
507:
501:
367:
362:
359:
354:
348:
343:
335:
331:
329:
326:Fine sheaves
317:
312:
285:
283:
264:
262:
257:
254:
245:
243:
117:
115:
98:
87:Tohoku paper
82:
81:in French),
78:
74:
70:
69:in French),
66:
62:
56:
35:
29:
18:Flabby sheaf
909:resolutions
907:, that is,
901:étalé space
784:, then the
351:paracompact
215:containing
46:(and other
32:mathematics
1064:Categories
936:References
878:surjective
668:subset of
635:soft sheaf
332:fine sheaf
97:notion of
1037:0040-8735
892:, etc.).
842:Γ
839:→
817:Γ
806:⊆
762:⊆
756:⊆
596:∙
484:→
476:
460:→
457:⋯
454:→
436:→
418:→
410:→
246:generator
984:(1998),
365:manifold
1045:0102537
1004:0345092
922:Flasque
890:modules
691:) is a
258:flasque
83:acyclic
67:flasque
59:sheaves
1043:
1035:
1002:
992:
965:
930:flabby
926:French
882:groups
666:closed
63:flabby
924:is a
886:rings
693:sheaf
660:over
334:over
311:over
89:" of
1033:ISSN
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71:fine
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473:dim
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284:An
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225:A
201:B
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