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''on a ringed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c0230201.png" />''
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{{MSC|14}}
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{{TEX|done}}
  
A sheaf of modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c0230202.png" /> over a sheaf of rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c0230203.png" /> with the following properties: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c0230204.png" /> is a sheaf of finite type, that is, it is locally generated over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c0230205.png" /> by a finite number of sections; and 2) the kernel of any homomorphism of sheaves of modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c0230206.png" /> over an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c0230207.png" /> is a sheaf of finite type. If in an exact sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c0230208.png" /> of sheaves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c0230209.png" />-modules two of the three sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302010.png" /> are coherent, then the third is coherent as well. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302011.png" /> is a homomorphism of coherent sheaves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302012.png" />-modules, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302015.png" /> are also coherent sheaves. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302017.png" /> are coherent, then so are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302019.png" /> [[#References|[4]]].
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''on a ringed space $(X,\def\cO{ {\mathcal O}}\cO)$''
  
A structure sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302020.png" /> is called a coherent sheaf of rings if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302021.png" /> is coherent as a sheaf of modules over itself, which reduces to condition 2). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302022.png" /> is a coherent sheaf of rings, then a sheaf of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302023.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302024.png" /> is coherent if and only if every point of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302025.png" /> has a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302026.png" /> over which there is an exact sequence of sheaves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302027.png" />-modules:
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A sheaf of modules $\def\cF{ {\mathcal F}}\cF$ over a sheaf of rings
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$\cO$ with the following properties:  
  
<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/c/c023/c023020/c02302028.png" /></td> </tr></table>
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1) $\cF$ is a sheaf of finite
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type, that is, it is locally generated over $\cO$ by a finite number
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of sections; and
  
[[#References|[4]]]. Furthermore, under this condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302029.png" /> is coherent for any coherent sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302031.png" /> and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302032.png" /> (see [[#References|[2]]]).
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2) the kernel of any homomorphism of sheaves of
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modules $\cO^p\mid_U\to \cF\mid_U$ over an open set $U\subset X$ is a sheaf of finite
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type.  
  
The fundamental classes of ringed spaces with a coherent structure sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302033.png" /> are: analytic spaces over algebraically closed fields [[#References|[1]]], Noetherian schemes and, in particular, algebraic varieties [[#References|[4]]]. A classical special case is the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302034.png" /> of germs of holomorphic functions in a domain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023020/c02302035.png" />; the statement that it is coherent is known as the Oka coherence theorem [[#References|[3]]], [[#References|[5]]]. The structure sheaf of a real-analytic space is not coherent, in general.
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If in an exact sequence $0\to \cF_1\to\cF_2\to\cF_3\to 0$ of sheaves of $\cO$-modules two of the three sheaves $\cF_i$ are coherent, then the third is coherent as well. If $\def\phi{\varphi}\phi:\cF\to\def\cS{ {\mathcal S}}\cS$ is a homomorphism of coherent sheaves of $\cO$-modules, then ${\rm Ker}\;\phi$, ${\rm Coker}\;\phi$, ${\rm Im}\;\phi$ are also coherent sheaves. If $\cF$ and $\cS$ are coherent, then so are $\cF\otimes_\cO \cS$ and ${\rm Hom}_\cO(\cF,\cS)$
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{{Cite|Se}}.
  
See also [[Coherent analytic sheaf|Coherent analytic sheaf]]; [[Coherent algebraic sheaf|Coherent algebraic sheaf]].
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A structure sheaf $\cO$ is called a coherent sheaf of rings if $\cO$ is coherent as a sheaf of modules over itself, which reduces to condition 2). If $\cO$ is a coherent sheaf of rings, then a sheaf of $\cO$-modules $\cF$ is coherent if and only if every point of the space $X$ has a neighbourhood $U$ over which there is an exact sequence of sheaves of $\cO$-modules:
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$$\cO^p\mid_U\to\cO^q\mid_U\to\cF\mid_U\to 0,$$
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{{Cite|Se}}. Furthermore, under this condition ${\rm Ext}_\cO^p(\cF,\cS)$ is coherent for any coherent sheaves $\cF$, $\cS$ and for all $p$ (see
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{{Cite|BaSt}}).
 +
 
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The fundamental classes of ringed spaces with a coherent structure sheaf $\cO$ are: analytic spaces over algebraically closed fields
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{{Cite|Ab}}, Noetherian schemes and, in particular, algebraic varieties
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{{Cite|Se}}. A classical special case is the sheaf $\cO$ of germs of holomorphic functions in a domain of $\mathbf C^n$; the statement that it is coherent is known as the Oka coherence theorem
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{{Cite|GuRo}},
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{{Cite|Fu}}. The structure sheaf of a real-analytic space is not coherent, in general.
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See also
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[[Coherent analytic sheaf|Coherent analytic sheaf]];
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[[Coherent algebraic sheaf|Coherent algebraic sheaf]].
  
 
====References====
 
====References====
<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"> 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">[4]</TD> <TD valign="top"> J.-P. Serre, "Faisceaux algébriques cohérents" ''Ann. of Math.'' , '''61''' (1955) pp. 197–278 {{MR|0068874}} {{ZBL|0067.16201}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> B.A. Fuks, "Special chapters in the theory of analytic functions of several complex variables" , Amer. Math. Soc. (1965) (Translated from Russian) {{MR|0188477}} {{ZBL|0146.30802}} </TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|Ab}}||valign="top"| S.S. Abhyankar, "Local analytic geometry", Acad. Press (1964) {{MR|0175897}} {{ZBL|0205.50401}}
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|-
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|valign="top"|{{Ref|BaSt}}||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}}
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|-
 +
|valign="top"|{{Ref|Fu}}||valign="top"| B.A. Fuks, "Special chapters in the theory of analytic functions of several complex variables", Amer. Math. Soc. (1965) (Translated from Russian) {{MR|0188477}} {{ZBL|0146.30802}}
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|-
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|valign="top"|{{Ref|GuRo}}||valign="top"| R.C. Gunning, H. Rossi, "Analytic functions of several complex variables", Prentice-Hall (1965) {{MR|0180696}} {{ZBL|0141.08601}}
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|-
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|valign="top"|{{Ref|Se}}||valign="top"| J.-P. Serre, "Faisceaux algébriques cohérents" ''Ann. of Math.'', '''61''' (1955) pp. 197–278 {{MR|0068874}} {{ZBL|0067.16201}}
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|-
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|}

Latest revision as of 13:34, 6 January 2022

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

on a ringed space $(X,\def\cO{ {\mathcal O}}\cO)$

A sheaf of modules $\def\cF{ {\mathcal F}}\cF$ over a sheaf of rings $\cO$ with the following properties:

1) $\cF$ is a sheaf of finite type, that is, it is locally generated over $\cO$ by a finite number of sections; and

2) the kernel of any homomorphism of sheaves of modules $\cO^p\mid_U\to \cF\mid_U$ over an open set $U\subset X$ is a sheaf of finite type.

If in an exact sequence $0\to \cF_1\to\cF_2\to\cF_3\to 0$ of sheaves of $\cO$-modules two of the three sheaves $\cF_i$ are coherent, then the third is coherent as well. If $\def\phi{\varphi}\phi:\cF\to\def\cS{ {\mathcal S}}\cS$ is a homomorphism of coherent sheaves of $\cO$-modules, then ${\rm Ker}\;\phi$, ${\rm Coker}\;\phi$, ${\rm Im}\;\phi$ are also coherent sheaves. If $\cF$ and $\cS$ are coherent, then so are $\cF\otimes_\cO \cS$ and ${\rm Hom}_\cO(\cF,\cS)$ [Se].

A structure sheaf $\cO$ is called a coherent sheaf of rings if $\cO$ is coherent as a sheaf of modules over itself, which reduces to condition 2). If $\cO$ is a coherent sheaf of rings, then a sheaf of $\cO$-modules $\cF$ is coherent if and only if every point of the space $X$ has a neighbourhood $U$ over which there is an exact sequence of sheaves of $\cO$-modules:

$$\cO^p\mid_U\to\cO^q\mid_U\to\cF\mid_U\to 0,$$ [Se]. Furthermore, under this condition ${\rm Ext}_\cO^p(\cF,\cS)$ is coherent for any coherent sheaves $\cF$, $\cS$ and for all $p$ (see [BaSt]).

The fundamental classes of ringed spaces with a coherent structure sheaf $\cO$ are: analytic spaces over algebraically closed fields [Ab], Noetherian schemes and, in particular, algebraic varieties [Se]. A classical special case is the sheaf $\cO$ of germs of holomorphic functions in a domain of $\mathbf C^n$; the statement that it is coherent is known as the Oka coherence theorem [GuRo], [Fu]. The structure sheaf of a real-analytic space is not coherent, in general.

See also Coherent analytic sheaf; Coherent algebraic sheaf.

References

[Ab] S.S. Abhyankar, "Local analytic geometry", Acad. Press (1964) MR0175897 Zbl 0205.50401
[BaSt] C. Banica, O. Stanasila, "Algebraic methods in the global theory of complex spaces", Wiley (1976) (Translated from Rumanian) MR0463470 Zbl 0334.32001
[Fu] B.A. Fuks, "Special chapters in the theory of analytic functions of several complex variables", Amer. Math. Soc. (1965) (Translated from Russian) MR0188477 Zbl 0146.30802
[GuRo] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables", Prentice-Hall (1965) MR0180696 Zbl 0141.08601
[Se] J.-P. Serre, "Faisceaux algébriques cohérents" Ann. of Math., 61 (1955) pp. 197–278 MR0068874 Zbl 0067.16201
How to Cite This Entry:
Coherent sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Coherent_sheaf&oldid=23789
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article