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A sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f0405401.png" /> of sets over a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f0405402.png" /> such that for any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f0405403.png" /> open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f0405404.png" /> the restriction mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f0405405.png" /> is surjective. Examples of such sheaves include the sheaf of germs of all (not necessarily continuous) sections of a [[Fibre space|fibre space]] with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f0405406.png" />, the sheaf of germs of divisors (cf. [[Divisor|Divisor]]), and a prime sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f0405407.png" /> over an irreducible algebraic variety. Flabbiness of a sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f0405408.png" /> is a local property (i.e. a flabby sheaf induces a flabby sheaf on any open set). A quotient sheaf of a flabby sheaf by a flabby sheaf is itself a flabby sheaf. The image of a flabby sheaf under a continuous mapping is a flabby sheaf. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f0405409.png" /> is paracompact, a flabby sheaf is a [[Soft sheaf|soft sheaf]], i.e. any section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f04054010.png" /> over a closed set can be extended to the entire space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f04054011.png" />.
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{{MSC|14}}
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{{TEX|done}}
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A ''flabby sheaf'' is
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a [[sheaf]] $F$ of sets over a topological space $X$ such that for any set $U$ open in $X$ the restriction mapping $F(X)\to F(U)$ is surjective. Examples of such sheaves include the sheaf of [[germ]]s of all (not necessarily continuous) [[section]]s of a
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[[Fibre space|fibre space]] with base $X$, the sheaf of germs of divisors (cf.
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[[Divisor (algebraic geometry)|Divisor]]), and a prime sheaf $F$ over an irreducible algebraic variety. Flabbiness of a sheaf $F$ is a local property (i.e. a flabby sheaf induces a flabby sheaf on any open set). A quotient sheaf of a flabby sheaf by a flabby sheaf is itself a flabby sheaf. The image of a flabby sheaf under a continuous mapping is a flabby sheaf. If $X$ is paracompact, a flabby sheaf is a
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[[Soft sheaf|soft sheaf]], i.e. any section of $F$ over a closed set can be extended to the entire space $X$.
  
 
Let
 
Let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f04054012.png" /></td> </tr></table>
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$$0\to F^0\to F^1\to \cdots$$
 
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be an exact sequence of flabby sheaves of Abelian groups. Then, for any family $\Phi$ of supports, the corresponding sequence of sections (the supports of which belong to $\Phi$)
be an exact sequence of flabby sheaves of Abelian groups. Then, for any family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f04054013.png" /> of supports, the corresponding sequence of sections (the supports of which belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f04054014.png" />)
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f04054015.png" /></td> </tr></table>
 
  
is exact, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040540/f04054016.png" /> is a left-exact functor.
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$$0\to\def\G{\Gamma}\G_\Phi(F^0)\to\G_\Phi(F^1)\to\cdots$$
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is exact, i.e. $F\mapsto \G_\Phi(F)$ is a left-exact functor.
  
  
  
 
====Comments====
 
====Comments====
Flabby sheaves are used for resolutions in the construction of sheaf cohomology (i.e. [[Cohomology|cohomology]] with values in a sheaf) in algebraic geometry and topology, [[#References|[a1]]].
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Flabby sheaves are used for resolutions in the construction of sheaf cohomology (i.e.
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[[Cohomology|cohomology]] with values in a sheaf) in algebraic geometry and topology,
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{{Cite|Mi}}.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.S. Milne, "Etale cohomology" , Princeton Univ. Press (1980) {{MR|0559531}} {{ZBL|0433.14012}} </TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|Mi}}||valign="top"| J.S. Milne, "Etale cohomology", Princeton Univ. Press (1980) {{MR|0559531}} {{ZBL|0433.14012}}
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|-
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Latest revision as of 18:47, 13 October 2014

2020 Mathematics Subject Classification: Primary: 14-XX [MSN][ZBL]

A flabby sheaf is a sheaf $F$ of sets over a topological space $X$ such that for any set $U$ open in $X$ the restriction mapping $F(X)\to F(U)$ is surjective. Examples of such sheaves include the sheaf of germs of all (not necessarily continuous) sections of a fibre space with base $X$, the sheaf of germs of divisors (cf. Divisor), and a prime sheaf $F$ over an irreducible algebraic variety. Flabbiness of a sheaf $F$ is a local property (i.e. a flabby sheaf induces a flabby sheaf on any open set). A quotient sheaf of a flabby sheaf by a flabby sheaf is itself a flabby sheaf. The image of a flabby sheaf under a continuous mapping is a flabby sheaf. If $X$ is paracompact, a flabby sheaf is a soft sheaf, i.e. any section of $F$ over a closed set can be extended to the entire space $X$.

Let

$$0\to F^0\to F^1\to \cdots$$ be an exact sequence of flabby sheaves of Abelian groups. Then, for any family $\Phi$ of supports, the corresponding sequence of sections (the supports of which belong to $\Phi$)

$$0\to\def\G{\Gamma}\G_\Phi(F^0)\to\G_\Phi(F^1)\to\cdots$$ is exact, i.e. $F\mapsto \G_\Phi(F)$ is a left-exact functor.


Comments

Flabby sheaves are used for resolutions in the construction of sheaf cohomology (i.e. cohomology with values in a sheaf) in algebraic geometry and topology, [Mi].

References

[Mi] J.S. Milne, "Etale cohomology", Princeton Univ. Press (1980) MR0559531 Zbl 0433.14012
How to Cite This Entry:
Flabby sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Flabby_sheaf&oldid=23831
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article