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Difference between revisions of "Analytic sheaf"

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A sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012420/a0124201.png" /> on an analytic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012420/a0124202.png" /> such that for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012420/a0124203.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012420/a0124204.png" /> is a [[Module|module]] over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012420/a0124205.png" /> of germs of holomorphic functions at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012420/a0124206.png" />, and such that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012420/a0124207.png" />, defined on the set of pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012420/a0124208.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012420/a0124209.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012420/a01242010.png" />, is a continuous mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012420/a01242011.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012420/a01242012.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012420/a01242013.png" />.
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A sheaf  $  F $
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on an analytic space  $  X $
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such that for any point  $  x \in X $
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the set  $  F _ {x} $
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is a [[Module|module]] over the ring  $  {\mathcal O} _ {x} $
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of germs of holomorphic functions at the point  $  x $,
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and such that the mapping  $  (f , \alpha ) \rightarrow f \alpha $,
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defined on the set of pairs  $  ( f, \alpha ) $
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where  $  f \in {\mathcal O} _ {x} $,
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$  \alpha \in F _ {x} $,
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is a continuous mapping of  $  {\mathcal O} \times F $
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into  $  F $
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for  $  x \in X $.
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Grauert,  R. Remmert,  "Theory of Stein spaces" , Springer  (1979)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Grauert,  R. Remmert,  "Coherent analytic sheaves" , Springer  (1984)  (Translated from German)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Grauert,  R. Remmert,  "Theory of Stein spaces" , Springer  (1979)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Grauert,  R. Remmert,  "Coherent analytic sheaves" , Springer  (1984)  (Translated from German)</TD></TR></table>

Latest revision as of 18:47, 5 April 2020


A sheaf $ F $ on an analytic space $ X $ such that for any point $ x \in X $ the set $ F _ {x} $ is a module over the ring $ {\mathcal O} _ {x} $ of germs of holomorphic functions at the point $ x $, and such that the mapping $ (f , \alpha ) \rightarrow f \alpha $, defined on the set of pairs $ ( f, \alpha ) $ where $ f \in {\mathcal O} _ {x} $, $ \alpha \in F _ {x} $, is a continuous mapping of $ {\mathcal O} \times F $ into $ F $ for $ x \in X $.

Comments

References

[a1] H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1979) (Translated from German)
[a2] H. Grauert, R. Remmert, "Coherent analytic sheaves" , Springer (1984) (Translated from German)
How to Cite This Entry:
Analytic sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_sheaf&oldid=18711
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article