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A coherent sheaf of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c0229901.png" /> modules on an [[Analytic space|analytic space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c0229902.png" />. A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c0229903.png" /> is said to be coherent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c0229904.png" /> is a coherent sheaf of rings. Any analytic space over an algebraically closed field is coherent. The most important examples of a coherent analytic sheaf on such a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c0229905.png" /> are a locally free sheaf (that is, an analytic sheaf locally isomorphic to the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c0229906.png" />) and also the sheaf of ideals of an analytic set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c0229907.png" />, that is, the sheaf of germs of analytic functions equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c0229908.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c0229909.png" />, [[#References|[1]]].
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A coherent sheaf of $\mathcal O$ modules on an [[Analytic space|analytic space]] $(X,\mathcal O)$. A space $(X,\mathcal O)$ is said to be coherent if $\mathcal O$ is a coherent sheaf of rings. Any analytic space over an algebraically closed field is coherent. The most important examples of a coherent analytic sheaf on such a space $(X,\mathcal O)$ are a locally free sheaf (that is, an analytic sheaf locally isomorphic to the sheaf $\mathcal O^p$) and also the sheaf of ideals of an analytic set $Y\subset X$, that is, the sheaf of germs of analytic functions equal to $0$ on $Y$, [[#References|[1]]].
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299010.png" /> is a coherent analytic sheaf on a complex-analytic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299011.png" />, then the space of its sections, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299012.png" />, is endowed with a natural topology turning it into a Fréchet space when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299013.png" /> is separable. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299014.png" />, this topology is the same as the topology of uniform convergence of analytic functions on compacta. In this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299015.png" /> becomes a Fréchet sheaf, that is, for arbitrary open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299016.png" /> the restriction mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299017.png" /> is continuous. An analytic homomorphism of coherent sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299018.png" /> induces a continuous linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299019.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299020.png" /> is a coherent analytic sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299022.png" /> is a submodule of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299024.png" />, then the submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299025.png" /> is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299026.png" /> for any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299028.png" />. The cohomology spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299029.png" /> also have a natural topology, which is not, in general, separable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299030.png" /> (they are quotient spaces of Fréchet spaces) [[#References|[2]]], [[#References|[4]]].
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If $\mathcal F$ is a coherent analytic sheaf on a complex-analytic space $(X,\mathcal O)$, then the space of its sections, $\Gamma(X,\mathcal F)$, is endowed with a natural topology turning it into a Fréchet space when $X$ is separable. For $\mathcal F=\mathcal O$, this topology is the same as the topology of uniform convergence of analytic functions on compacta. In this case, $\mathcal F$ becomes a Fréchet sheaf, that is, for arbitrary open sets $U\subset V\subset X$ the restriction mapping $\Gamma(V,\mathcal F)\to\Gamma(U,\mathcal F)$ is continuous. An analytic homomorphism of coherent sheaves $\mathcal F\to\mathcal G$ induces a continuous linear mapping $\Gamma(X,\mathcal F)\to\Gamma(X,\mathcal G)$. If $\mathcal F$ is a coherent analytic sheaf on $X$ and $M$ is a submodule of $\mathcal F_x$, $x\in X$, then the submodule $\{s\in\Gamma(U,\mathcal F)\colon s(x)\in M\}$ is closed in $\Gamma(U,\mathcal F)$ for any neighbourhood $U$ of $x$. The cohomology spaces $H^p(X,\mathcal F)$ also have a natural topology, which is not, in general, separable for $p>0$ (they are quotient spaces of Fréchet spaces) [[#References|[2]]], [[#References|[4]]].
  
Coherent analytic sheaves were introduced in connection with problems in the theory of analytic functions on domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022990/c02299031.png" /> (see [[#References|[3]]], [[#References|[5]]]). Later they and their cohomology became a fundamental tool in the global theory of analytic spaces. Criteria for the vanishing of cohomology with values in a coherent analytic sheaf (cf. [[Kodaira theorem|Kodaira theorem]]; [[Ample vector bundle|Ample vector bundle]]; [[Stein space|Stein space]]) as well as criteria for its finiteness and separability (see [[Finiteness theorems|Finiteness theorems]] in the theory of analytic spaces) play an important role in this theory.
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Coherent analytic sheaves were introduced in connection with problems in the theory of analytic functions on domains in $\mathbf C^n$ (see [[#References|[3]]], [[#References|[5]]]). Later they and their cohomology became a fundamental tool in the global theory of analytic spaces. Criteria for the vanishing of cohomology with values in a coherent analytic sheaf (cf. [[Kodaira theorem|Kodaira theorem]]; [[Ample vector bundle|Ample vector bundle]]; [[Stein space|Stein space]]) as well as criteria for its finiteness and separability (see [[Finiteness theorems|Finiteness theorems]] in the theory of analytic spaces) play an important role in this theory.
  
 
See also [[Vector bundle, analytic|Vector bundle, analytic]]; [[Duality|Duality]] in the theory of analytic spaces.
 
See also [[Vector bundle, analytic|Vector bundle, analytic]]; [[Duality|Duality]] in the theory of analytic spaces.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.S. Abhyankar,   "Local analytic geometry" , Acad. Press (1964)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> C. Banica,   O. Stanasila,   "Algebraic methods in the global theory of complex spaces" , Wiley (1976) (Translated from Rumanian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> H. Cartan,   "Idéaux et modules de fonctions analytiques de variables complexes" ''Bull. Soc. Math. France'' , '''78''' (1950) pp. 28–64</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R.C. Gunning,   H. Rossi,   "Analytic functions of several complex variables" , Prentice-Hall (1965)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> K. Oka,   "Sur les fonctions analytiques de plusieurs variables (VII. Sur quelques notions arithmétiques)" ''Bull. Soc. Math. France'' , '''78''' (1950) pp. 1–27</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.S. Abhyankar, "Local analytic geometry" , Acad. Press (1964) {{MR|0175897}} {{ZBL|0205.50401}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> C. Banica, O. Stanasila, "Algebraic methods in the global theory of complex spaces" , Wiley (1976) (Translated from Rumanian) {{MR|0463470}} {{ZBL|0334.32001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> H. Cartan, "Idéaux et modules de fonctions analytiques de variables complexes" ''Bull. Soc. Math. France'' , '''78''' (1950) pp. 28–64 {{MR|0036848}} {{ZBL|0038.23703}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) {{MR|0180696}} {{ZBL|0141.08601}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> K. Oka, "Sur les fonctions analytiques de plusieurs variables (VII. Sur quelques notions arithmétiques)" ''Bull. Soc. Math. France'' , '''78''' (1950) pp. 1–27 {{MR|0035831}} {{ZBL|0036.05202}} </TD></TR></table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Grauert,   R. Remmert,   "Coherent analytic sheaves" , Springer (1984) (Translated from German)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Grauert, R. Remmert, "Coherent analytic sheaves" , Springer (1984) (Translated from German) {{MR|0755331}} {{ZBL|0537.32001}} </TD></TR></table>

Latest revision as of 08:18, 22 August 2014

A coherent sheaf of $\mathcal O$ modules on an analytic space $(X,\mathcal O)$. A space $(X,\mathcal O)$ is said to be coherent if $\mathcal O$ is a coherent sheaf of rings. Any analytic space over an algebraically closed field is coherent. The most important examples of a coherent analytic sheaf on such a space $(X,\mathcal O)$ are a locally free sheaf (that is, an analytic sheaf locally isomorphic to the sheaf $\mathcal O^p$) and also the sheaf of ideals of an analytic set $Y\subset X$, that is, the sheaf of germs of analytic functions equal to $0$ on $Y$, [1].

If $\mathcal F$ is a coherent analytic sheaf on a complex-analytic space $(X,\mathcal O)$, then the space of its sections, $\Gamma(X,\mathcal F)$, is endowed with a natural topology turning it into a Fréchet space when $X$ is separable. For $\mathcal F=\mathcal O$, this topology is the same as the topology of uniform convergence of analytic functions on compacta. In this case, $\mathcal F$ becomes a Fréchet sheaf, that is, for arbitrary open sets $U\subset V\subset X$ the restriction mapping $\Gamma(V,\mathcal F)\to\Gamma(U,\mathcal F)$ is continuous. An analytic homomorphism of coherent sheaves $\mathcal F\to\mathcal G$ induces a continuous linear mapping $\Gamma(X,\mathcal F)\to\Gamma(X,\mathcal G)$. If $\mathcal F$ is a coherent analytic sheaf on $X$ and $M$ is a submodule of $\mathcal F_x$, $x\in X$, then the submodule $\{s\in\Gamma(U,\mathcal F)\colon s(x)\in M\}$ is closed in $\Gamma(U,\mathcal F)$ for any neighbourhood $U$ of $x$. The cohomology spaces $H^p(X,\mathcal F)$ also have a natural topology, which is not, in general, separable for $p>0$ (they are quotient spaces of Fréchet spaces) [2], [4].

Coherent analytic sheaves were introduced in connection with problems in the theory of analytic functions on domains in $\mathbf C^n$ (see [3], [5]). Later they and their cohomology became a fundamental tool in the global theory of analytic spaces. Criteria for the vanishing of cohomology with values in a coherent analytic sheaf (cf. Kodaira theorem; Ample vector bundle; Stein space) as well as criteria for its finiteness and separability (see Finiteness theorems in the theory of analytic spaces) play an important role in this theory.

See also Vector bundle, analytic; Duality in the theory of analytic spaces.

References

[1] S.S. Abhyankar, "Local analytic geometry" , Acad. Press (1964) MR0175897 Zbl 0205.50401
[2] C. Banica, O. Stanasila, "Algebraic methods in the global theory of complex spaces" , Wiley (1976) (Translated from Rumanian) MR0463470 Zbl 0334.32001
[3] H. Cartan, "Idéaux et modules de fonctions analytiques de variables complexes" Bull. Soc. Math. France , 78 (1950) pp. 28–64 MR0036848 Zbl 0038.23703
[4] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) MR0180696 Zbl 0141.08601
[5] K. Oka, "Sur les fonctions analytiques de plusieurs variables (VII. Sur quelques notions arithmétiques)" Bull. Soc. Math. France , 78 (1950) pp. 1–27 MR0035831 Zbl 0036.05202


Comments

See also Coherent sheaf.

References

[a1] H. Grauert, R. Remmert, "Coherent analytic sheaves" , Springer (1984) (Translated from German) MR0755331 Zbl 0537.32001
How to Cite This Entry:
Coherent analytic sheaf. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Coherent_analytic_sheaf&oldid=18834
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article