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Difference between revisions of "Fine sheaf"

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A sheaf of Abelian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040180/f0401801.png" /> over a paracompact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040180/f0401802.png" /> with a [[Soft sheaf|soft sheaf]] as sheaf of endomorphisms. A sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040180/f0401803.png" /> is fine if and only if for any closed subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040180/f0401804.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040180/f0401805.png" /> there is an endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040180/f0401806.png" /> that is the identity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040180/f0401807.png" /> and zero on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040180/f0401808.png" />, or equivalently if for every open covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040180/f0401809.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040180/f04018010.png" /> there is a locally finite collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040180/f04018011.png" /> of endomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040180/f04018012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040180/f04018013.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040180/f04018014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040180/f04018015.png" /> is the identity endomorphism. Every fine sheaf is soft, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040180/f04018016.png" /> is a sheaf of rings with an identity, the converse also holds. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040180/f04018017.png" /> is a fine sheaf and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040180/f04018018.png" /> is an arbitrary sheaf of Abelian groups, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040180/f04018019.png" /> is also a fine sheaf. An example of a fine sheaf is the sheaf of germs of continuous (or differentiable of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040180/f04018020.png" />) sections of a vector bundle over a paracompact space (respectively, over a paracompact differentiable manifold).
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A sheaf of Abelian groups $\mathcal F$ over a paracompact space $X$ with a [[Soft sheaf|soft sheaf]] as sheaf of endomorphisms. A sheaf $\mathcal F$ is fine if and only if for any closed subsets $A,B\subset X$ with $A\cap B=\emptyset$ there is an endomorphism $h\colon\mathcal F\to\mathcal F$ that is the identity on $A$ and zero on $B$, or equivalently if for every open covering $(U_i)_{i\in I}$ of $X$ there is a locally finite collection $(h_i)_{i\in I}$ of endomorphisms of $\mathcal F$ such that $\supp h_i\subset U_i$ $(i\in I)$ and $\sum_{i\in I}h_i$ is the identity endomorphism. Every fine sheaf is soft, and if $\mathcal F$ is a sheaf of rings with an identity, the converse also holds. If $\mathcal F$ is a fine sheaf and $\mathcal L$ is an arbitrary sheaf of Abelian groups, then $\mathcal F\otimes_\mathbf Z\mathcal L$ is also a fine sheaf. An example of a fine sheaf is the sheaf of germs of continuous (or differentiable of class $C^k$) sections of a vector bundle over a paracompact space (respectively, over a paracompact differentiable manifold).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Godement,  "Topologie algébrique et théorie des faisceaux" , Hermann  (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.O. Wells jr.,  "Differential analysis on complex manifolds" , Springer  (1980)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Godement,  "Topologie algébrique et théorie des faisceaux" , Hermann  (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.O. Wells jr.,  "Differential analysis on complex manifolds" , Springer  (1980)</TD></TR></table>

Latest revision as of 13:05, 29 November 2018

A sheaf of Abelian groups $\mathcal F$ over a paracompact space $X$ with a soft sheaf as sheaf of endomorphisms. A sheaf $\mathcal F$ is fine if and only if for any closed subsets $A,B\subset X$ with $A\cap B=\emptyset$ there is an endomorphism $h\colon\mathcal F\to\mathcal F$ that is the identity on $A$ and zero on $B$, or equivalently if for every open covering $(U_i)_{i\in I}$ of $X$ there is a locally finite collection $(h_i)_{i\in I}$ of endomorphisms of $\mathcal F$ such that $\supp h_i\subset U_i$ $(i\in I)$ and $\sum_{i\in I}h_i$ is the identity endomorphism. Every fine sheaf is soft, and if $\mathcal F$ is a sheaf of rings with an identity, the converse also holds. If $\mathcal F$ is a fine sheaf and $\mathcal L$ is an arbitrary sheaf of Abelian groups, then $\mathcal F\otimes_\mathbf Z\mathcal L$ is also a fine sheaf. An example of a fine sheaf is the sheaf of germs of continuous (or differentiable of class $C^k$) sections of a vector bundle over a paracompact space (respectively, over a paracompact differentiable manifold).

References

[1] R. Godement, "Topologie algébrique et théorie des faisceaux" , Hermann (1958)
[2] R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)
How to Cite This Entry:
Fine sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fine_sheaf&oldid=43510
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article