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A [[Sheaf|sheaf]] of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s0860001.png" /> on a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s0860002.png" /> any section of which over some closed subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s0860003.png" /> can be extended to a section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s0860004.png" /> over all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s0860005.png" />. Examples of soft sheaves are: the sheaf of germs of discontinuous sections of an arbitrary sheaf of sets on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s0860006.png" />; any [[Flabby sheaf|flabby sheaf]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s0860007.png" /> on a [[Paracompact space|paracompact space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s0860008.png" />; and any [[Fine sheaf|fine sheaf]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s0860009.png" /> of Abelian groups on a paracompact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s08600010.png" />. The property of softness of a sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s08600011.png" /> on a paracompact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s08600012.png" /> is local: A sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s08600013.png" /> is soft if and only if any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s08600014.png" /> has an open neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s08600015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s08600016.png" /> is a soft sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s08600017.png" />. A soft sheaf on a paracompact space induces a soft sheaf on any closed (and, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s08600018.png" /> is metrizable, any locally closed) subspace. A sheaf of modules over a soft sheaf of rings is a soft sheaf.
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A [[Sheaf|sheaf]] of sets $\mathcal F$ on a topological space $X$ any section of which over some closed subset in $X$ can be extended to a section of $\mathcal F$ over all of $X$. Examples of soft sheaves are: the sheaf of germs of discontinuous sections of an arbitrary sheaf of sets on $X$; any [[Flabby sheaf|flabby sheaf]] $\mathcal F$ on a [[Paracompact space|paracompact space]] $X$; and any [[Fine sheaf|fine sheaf]] $\mathcal F$ of Abelian groups on a paracompact space $X$. The property of softness of a sheaf $\mathcal F$ on a paracompact space $X$ is local: A sheaf $\mathcal F$ is soft if and only if any $x\in X$ has an open neighbourhood $U$ such that $\mathcal F|_U$ is a soft sheaf on $U$. A soft sheaf on a paracompact space induces a soft sheaf on any closed (and, if $X$ is metrizable, any locally closed) subspace. A sheaf of modules over a soft sheaf of rings is a soft sheaf.
  
 
If
 
If
  
<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/s/s086/s086000/s08600019.png" /></td> </tr></table>
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$$0\to\mathcal F^0\to\mathcal F^1\to\dots$$
  
is an exact sequence of soft sheaves of Abelian groups on a paracompact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s08600020.png" />, then the corresponding sequence of groups of sections
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is an exact sequence of soft sheaves of Abelian groups on a paracompact space $X$, then the corresponding sequence of groups of sections
  
<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/s/s086/s086000/s08600021.png" /></td> </tr></table>
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$$0\to\mathcal F^0(X)\to\mathcal F^1(X)\to\dots$$
  
is also exact. The cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s08600022.png" /> of any soft sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s08600023.png" /> of Abelian groups on a paracompact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s08600024.png" /> is trivial for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086000/s08600025.png" />.
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is also exact. The cohomology group $H^p(X,\mathcal F)$ of any soft sheaf $\mathcal F$ of Abelian groups on a paracompact space $X$ is trivial for $p>0$.
  
 
====References====
 
====References====

Revision as of 12:58, 29 November 2018

A sheaf of sets $\mathcal F$ on a topological space $X$ any section of which over some closed subset in $X$ can be extended to a section of $\mathcal F$ over all of $X$. Examples of soft sheaves are: the sheaf of germs of discontinuous sections of an arbitrary sheaf of sets on $X$; any flabby sheaf $\mathcal F$ on a paracompact space $X$; and any fine sheaf $\mathcal F$ of Abelian groups on a paracompact space $X$. The property of softness of a sheaf $\mathcal F$ on a paracompact space $X$ is local: A sheaf $\mathcal F$ is soft if and only if any $x\in X$ has an open neighbourhood $U$ such that $\mathcal F|_U$ is a soft sheaf on $U$. A soft sheaf on a paracompact space induces a soft sheaf on any closed (and, if $X$ is metrizable, any locally closed) subspace. A sheaf of modules over a soft sheaf of rings is a soft sheaf.

If

$$0\to\mathcal F^0\to\mathcal F^1\to\dots$$

is an exact sequence of soft sheaves of Abelian groups on a paracompact space $X$, then the corresponding sequence of groups of sections

$$0\to\mathcal F^0(X)\to\mathcal F^1(X)\to\dots$$

is also exact. The cohomology group $H^p(X,\mathcal F)$ of any soft sheaf $\mathcal F$ of Abelian groups on a paracompact space $X$ is trivial for $p>0$.

References

[1] R. Godement, "Topologie algébrique et théorie des faisceaux" , Hermann (1958)
[2] R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)


Comments

References

[a1] G.E. Bredon, "Sheaf theory" , McGraw-Hill (1967) pp. §9
How to Cite This Entry:
Soft sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Soft_sheaf&oldid=43509
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article