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''with values in a sheaf of Abelian groups''
 
''with values in a sheaf of Abelian groups''
  
A [[Cohomology|cohomology]] theory with values in a [[Sheaf|sheaf]] and with supports contained in a given subset. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l0600901.png" /> be a [[Topological space|topological space]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l0600902.png" /> a sheaf of Abelian groups on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l0600903.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l0600904.png" /> a locally closed subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l0600905.png" />, that is, a closed subset of some subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l0600906.png" /> open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l0600907.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l0600908.png" /> denotes the subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l0600909.png" /> consisting of the sections of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009010.png" /> with supports in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009011.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009012.png" /> is fixed, then the correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009013.png" /> defines a left-exact functor from the category of sheaves of Abelian groups on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009014.png" /> into the category of Abelian groups. The value of the corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009015.png" />-th right [[Derived functor|derived functor]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009016.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009017.png" /> and is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009018.png" />-th local cohomology group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009019.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009020.png" />, with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009021.png" />. One has
+
A [[Cohomology|cohomology]] theory with values in a [[Sheaf|sheaf]] and with supports contained in a given subset. Let $  X $
 +
be a [[Topological space|topological space]], $  {\mathcal F} $
 +
a sheaf of Abelian groups on $  X $
 +
and $  Z $
 +
a locally closed subset of $  X $,  
 +
that is, a closed subset of some subset $  V $
 +
open in $  X $.  
 +
Then $  \Gamma _ {Z} ( X , {\mathcal F} ) $
 +
denotes the subgroup of $  \Gamma ( V , {\mathcal F} \mid  _ {V} ) $
 +
consisting of the sections of the sheaf $  {\mathcal F} \mid  _ {V} $
 +
with supports in $  Z $.  
 +
If $  Z $
 +
is fixed, then the correspondence $  {\mathcal F} \rightarrow \Gamma _ {Z} ( X , {\mathcal F} ) $
 +
defines a left-exact functor from the category of sheaves of Abelian groups on $  X $
 +
into the category of Abelian groups. The value of the corresponding $  i $-
 +
th right [[Derived functor|derived functor]] on $  {\mathcal F} $
 +
is denoted by $  H _ {Z}  ^ {i} ( X , {\mathcal F} ) $
 +
and is called the $  i $-
 +
th local cohomology group of $  X $
 +
with values in $  {\mathcal F} $,  
 +
with respect to $  Z $.  
 +
One has
  
<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/l/l060/l060090/l06009022.png" /></td> </tr></table>
+
$$
 +
H _ {Z}  ^ {0} ( X , {\mathcal F} )  = \Gamma _ {Z} ( X , {\mathcal F} ) .
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009023.png" /> be the sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009024.png" /> corresponding to the pre-sheaf that associates with any open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009025.png" /> the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009026.png" />. The correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009027.png" /> is a left-exact functor from the category of sheaves of Abelian groups on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009028.png" /> into itself. The value of its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009029.png" />-th right derived functor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009030.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009031.png" /> and is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009032.png" />-th local cohomology sheaf of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009033.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009034.png" />. The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009035.png" /> is associated with the pre-sheaf that associates with an open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009036.png" /> the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009037.png" />.
+
Let $  {\mathcal H} _ {Z}  ^ {0} ( {\mathcal F} ) $
 +
be the sheaf on $  X $
 +
corresponding to the pre-sheaf that associates with any open subset $  U \subset  X $
 +
the group $  \Gamma _ {Z \cap U }  ( U , {\mathcal F} \mid  _ {U} ) $.  
 +
The correspondence $  {\mathcal F} \rightarrow {\mathcal H} _ {Z} ( {\mathcal F} ) $
 +
is a left-exact functor from the category of sheaves of Abelian groups on $  X $
 +
into itself. The value of its $  i $-
 +
th right derived functor on $  {\mathcal F} $
 +
is denoted by $  {\mathcal H} _ {Z} ( {\mathcal F} ) $
 +
and is called the $  i $-
 +
th local cohomology sheaf of $  {\mathcal F} $
 +
with respect to $  Z $.  
 +
The sheaf $  {\mathcal H} _ {Z}  ^ {i} ( {\mathcal F} ) $
 +
is associated with the pre-sheaf that associates with an open subset $  U \subset  X $
 +
the group $  H _ {Z \cap U }  ^ {i} ( U , {\mathcal F} \mid  _ {U} ) $.
  
There is a spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009038.png" />, converging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009039.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009040.png" /> (see [[#References|[2]]], [[#References|[3]]]).
+
There is a spectral sequence $  E _ {r}  ^ {p,q} $,  
 +
converging to $  H _ {Z}  ^ {p+} q ( X , {\mathcal F} ) $,  
 +
for which $  E _ {2}  ^ {p,q} = H  ^ {p} ( X , {\mathcal H} _ {Z}  ^ {q} ( {\mathcal F} ) ) $(
 +
see [[#References|[2]]], [[#References|[3]]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009041.png" /> be a locally closed subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009043.png" /> a closed subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009045.png" />; then there are the following exact sequences:
+
Let $  Z $
 +
be a locally closed subset of $  X $,  
 +
$  Z  ^  \prime  $
 +
a closed subset of $  Z $
 +
and $  Z  ^ {\prime\prime} = Z \setminus  Z  ^  \prime  $;  
 +
then there are the following exact sequences:
  
<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/l/l060/l060090/l06009046.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
0 \rightarrow  H _ {Z  ^  \prime  }  ^ {0} ( X , {\mathcal F} )  \rightarrow \dots \rightarrow \
 +
H _ {Z  ^  \prime  }  ^ {i} ( X , {\mathcal F} ) \rightarrow
 +
$$
  
<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/l/l060/l060090/l06009047.png" /></td> </tr></table>
+
$$
 +
\rightarrow \
 +
H _ {Z}  ^ {i} ( X , {\mathcal F} )  \rightarrow  H _ {Z  ^ {\prime\prime}  }  ^ {i} (
 +
X , {\mathcal F} )  \rightarrow  H _ {Z  ^  \prime  }  ^ {i+} 1 ( X , {\mathcal F} )  \rightarrow \dots ;
 +
$$
  
<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/l/l060/l060090/l06009048.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
0 \rightarrow  {\mathcal H} _ {Z  ^  \prime  }  ^ {0} ( {\mathcal F} )  \rightarrow
 +
\dots \rightarrow  {\mathcal H} _ {Z  ^  \prime  }  ^ {i} ( {\mathcal F} ) \rightarrow
 +
$$
  
<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/l/l060/l060090/l06009049.png" /></td> </tr></table>
+
$$
 +
\rightarrow \
 +
{\mathcal H} _ {Z}  ^ {i} ( {\mathcal F} )  \rightarrow  {\mathcal H} _ {Z  ^ {\prime\prime}  }  ^ {i}
 +
( {\mathcal F} )  \rightarrow  {\mathcal H} _ {Z  ^  \prime  }  ^ {i+} 1 ( {\mathcal F} )  \rightarrow \dots .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009050.png" /> is the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009052.png" /> is a closed subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009053.png" />, then the sequence (2) gives the exact sequence
+
If $  Z $
 +
is the whole of $  X $
 +
and $  Z  ^  \prime  $
 +
is a closed subset of $  X $,  
 +
then the sequence (2) gives the exact sequence
  
<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/l/l060/l060090/l06009054.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow  {\mathcal H} _ {Z  ^  \prime  }  ^ {0} ( {\mathcal F} )  \rightarrow  {\mathcal F}  \rightarrow \
 +
{\mathcal H} _ {X \setminus  Z  ^  \prime  }  ^ {0} ( {\mathcal F} )  \rightarrow  {\mathcal H} _ {Z  ^  \prime  }
 +
^ {1} ( {\mathcal F} )  \rightarrow  0
 +
$$
  
 
and the system of isomorphisms
 
and the system of isomorphisms
  
<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/l/l060/l060090/l06009055.png" /></td> </tr></table>
+
$$
 +
{\mathcal H} _ {X \setminus  Z  ^  \prime  }  ^ {i} ( {\mathcal F} )  \cong \
 +
{\mathcal H} _ {Z  ^  \prime  }  ^ {i+} 1 ( {\mathcal F} ) ,\  i \geq  1 .
 +
$$
  
The sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009056.png" /> are called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009057.png" />-th gap sheaves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009058.png" /> and have important applications in questions concerning the extension of sections and cohomology classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009059.png" />, defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009060.png" />, to the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009061.png" /> (see [[#References|[4]]]).
+
The sheaves $  {\mathcal H} _ {X \setminus  Z  ^  \prime  }  ^ {i} ( {\mathcal F} ) $
 +
are called the $  i $-
 +
th gap sheaves of $  {\mathcal F} $
 +
and have important applications in questions concerning the extension of sections and cohomology classes of $  {\mathcal F} $,  
 +
defined on $  X \setminus  Z  ^  \prime  $,  
 +
to the whole of $  X $(
 +
see [[#References|[4]]]).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009062.png" /> is a locally [[Noetherian scheme|Noetherian scheme]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009063.png" /> is a [[Quasi-coherent sheaf|quasi-coherent sheaf]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009065.png" /> is a closed subscheme of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009066.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009067.png" /> are quasi-coherent sheaves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009068.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009069.png" /> is a [[Coherent sheaf|coherent sheaf]] of ideals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009070.png" /> that specifies the subscheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009071.png" />, then one has the isomorphisms
+
If $  X $
 +
is a locally [[Noetherian scheme|Noetherian scheme]], $  {\mathcal F} $
 +
is a [[Quasi-coherent sheaf|quasi-coherent sheaf]] on $  X $
 +
and $  Z $
 +
is a closed subscheme of $  X $,  
 +
then $  {\mathcal H} _ {Z}  ^ {i} ( {\mathcal F} ) $
 +
are quasi-coherent sheaves on $  X $.  
 +
If $  {\mathcal Y} $
 +
is a [[Coherent sheaf|coherent sheaf]] of ideals on $  X $
 +
that specifies the subscheme $  Z $,  
 +
then one has the isomorphisms
  
<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/l/l060/l060090/l06009072.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ { {\  n \  } vec } \
 +
\mathop{\rm Ext} _ { {\mathcal O} _ {X}  }  ^ {i}
 +
( {\mathcal O} _ {X} / {\mathcal Y}  ^ {n} , {\mathcal F} )  \cong \
 +
{\mathcal H} _ {Z}  ^ {i} ( {\mathcal F} ) .
 +
$$
  
 
The following criteria for triviality and coherence of local cohomology sheaves are important for applications (see [[#References|[3]]], [[#References|[4]]]).
 
The following criteria for triviality and coherence of local cohomology sheaves are important for applications (see [[#References|[3]]], [[#References|[4]]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009073.png" /> be a locally Noetherian scheme or a complex-analytic space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009074.png" /> a locally closed subscheme or analytic subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009076.png" /> a coherent sheaf of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009077.png" />-modules, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009078.png" /> a coherent sheaf of ideals that specifies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009079.png" />. Let
+
Let $  X $
 +
be a locally Noetherian scheme or a complex-analytic space, $  Z $
 +
a locally closed subscheme or analytic subspace of $  X $,  
 +
$  {\mathcal F} $
 +
a coherent sheaf of $  {\mathcal O} _ {X} $-
 +
modules, and $  {\mathcal Y} $
 +
a coherent sheaf of ideals that specifies $  Z $.  
 +
Let
  
<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/l/l060/l060090/l06009080.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm prof} _ {Z}  {\mathcal F}  = \
 +
\min _ {x \in Z } \
 +
\mathop{\rm prof} _ { {\mathcal Y} _ {X,x} }  {\mathcal F} _ {x} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009081.png" /> is the maximal length of a sequence of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009082.png" /> that is regular for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009083.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009084.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009085.png" />. Then the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009086.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009087.png" /> is equivalent to the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009088.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009089.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009090.png" /> is the maximal ideal of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009091.png" />) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009092.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009093.png" /> is a complex-analytic space or an algebraic variety, then all sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009094.png" /> are analytic or algebraic, respectively. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009095.png" /> is a coherent sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009097.png" /> is an analytic subspace or subvariety, respectively, then coherence of the sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009098.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l06009099.png" /> is equivalent to the condition
+
where $  \mathop{\rm prof} _ { {\mathcal Y} _ {X,x} }  {\mathcal F} _ {x} $
 +
is the maximal length of a sequence of elements of $  {\mathcal Y} _ {X,x} $
 +
that is regular for $  {\mathcal F} _ {x} $,  
 +
or $  \infty $
 +
if $  {\mathcal F} _ {x} = 0 $.  
 +
Then the equality $  {\mathcal H} _ {Z}  ^ {i} ( {\mathcal F} ) = 0 $
 +
for $  i < n $
 +
is equivalent to the condition $  \mathop{\rm prof} _ {Z}  {\mathcal F} \geq  n $.  
 +
Let $  \mathop{\rm codh} _ {x}  {\mathcal F} _ {x} = \mathop{\rm prof} _ {\mathfrak m _ {x}  }  {\mathcal F} _ {x} $(
 +
where $  \mathfrak m $
 +
is the maximal ideal of the ring $  {\mathcal O} _ {X,x} $)  
 +
and let $  S _ {m} ( {\mathcal F} ) = \{ {x \in X } : { \mathop{\rm codh} _ {x}  {\mathcal F} _ {x} \geq  m } \} $.  
 +
If $  X $
 +
is a complex-analytic space or an algebraic variety, then all sets $  S _ {m} ( {\mathcal F} ) $
 +
are analytic or algebraic, respectively. If $  {\mathcal F} $
 +
is a coherent sheaf on $  X $
 +
and $  Z $
 +
is an analytic subspace or subvariety, respectively, then coherence of the sheaves $  {\mathcal H} _ {Z}  ^ {i} ( {\mathcal F} ) $
 +
for 0 \leq  i \leq  q $
 +
is equivalent to the condition
  
<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/l/l060/l060090/l060090100.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm dim}  Z \cap \overline{ {S _ {k+} q+ 1 }}\;
 +
( {\mathcal F}  \mid  _ {X \setminus  Z }  )  \leq  k
 +
$$
  
for any integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090101.png" />.
+
for any integer $  k $.
  
In terms of local cohomology one can define hyperfunctions, which have important applications in the theory of partial differential equations [[#References|[5]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090102.png" /> be an open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090103.png" />, which is naturally imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090104.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090105.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090106.png" />. The pre-sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090107.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090108.png" /> defines a [[Flabby sheaf|flabby sheaf]], called the sheaf of hyperfunctions.
+
In terms of local cohomology one can define hyperfunctions, which have important applications in the theory of partial differential equations [[#References|[5]]]. Let $  \Omega $
 +
be an open subset of $  \mathbf R  ^ {n} $,  
 +
which is naturally imbedded in $  \mathbf C  ^ {n} $.  
 +
Then $  {\mathcal H} _  \Omega  ^ {p} ( \mathbf C  ^ {n} , {\mathcal O} _ {\mathbf C  ^ {n}  } ) = 0 $
 +
for $  p \neq n $.  
 +
The pre-sheaf $  \Omega \rightarrow {\mathcal H} _  \Omega  ^ {n} ( \mathbf C  ^ {n} , {\mathcal O} _ {\mathbf C  ^ {n}  } ) $
 +
on $  \mathbf R  ^ {n} $
 +
defines a [[Flabby sheaf|flabby sheaf]], called the sheaf of hyperfunctions.
  
 
An analogue of local cohomology also exists in [[Etale cohomology|étale cohomology]] theory [[#References|[3]]].
 
An analogue of local cohomology also exists in [[Etale cohomology|étale cohomology]] theory [[#References|[3]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.V. Dolgachev,   "Abstract algebraic geometry" ''Russian Math. Surveys'' , '''2''' : 3 (1974) pp. 264–303 ''Itogi Nauk. i Tekhn. Algebra. Topol. Geom.'' , '''10''' (1972) pp. 47–112</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Grothendieck,   "Local cohomology" , ''Lect. notes in math.'' , '''41''' , Springer (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Grothendieck,   "Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux" , ''SGA 2'' , North-Holland &amp; Masson (1968)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> Y.-T. Siu,   "Gap-sheaves and extension of coherent analytic subsheaves" , Springer (1971)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> P. Schapira,   "Théorie des hyperfonctions" , ''Lect. notes in math.'' , '''126''' , Springer (1970)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> C. Banica,   O. Stanasila,   "Algebraic methods in the global theory of complex spaces" , Wiley (1976) (Translated from Rumanian)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.V. Dolgachev, "Abstract algebraic geometry" ''Russian Math. Surveys'' , '''2''' : 3 (1974) pp. 264–303 ''Itogi Nauk. i Tekhn. Algebra. Topol. Geom.'' , '''10''' (1972) pp. 47–112 {{MR|}} {{ZBL|1068.14059}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Grothendieck, "Local cohomology" , ''Lect. notes in math.'' , '''41''' , Springer (1967) {{MR|0224620}} {{ZBL|0185.49202}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Grothendieck, "Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux" , ''SGA 2'' , North-Holland &amp; Masson (1968) {{MR|0476737}} {{ZBL|1079.14001}} {{ZBL|0159.50402}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> Y.-T. Siu, "Gap-sheaves and extension of coherent analytic subsheaves" , Springer (1971) {{MR|0287033}} {{ZBL|0208.10403}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> P. Schapira, "Théorie des hyperfonctions" , ''Lect. notes in math.'' , '''126''' , Springer (1970) {{MR|0631543}} {{MR|0270151}} {{ZBL|0201.44805}} {{ZBL|0192.47305}} </TD></TR><TR><TD valign="top">[6]</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></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
 
See also [[Hyperfunction|Hyperfunction]] for the sheaf of hyperfunctions.
 
See also [[Hyperfunction|Hyperfunction]] for the sheaf of hyperfunctions.
  
For an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090109.png" /> in a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090110.png" /> with unit element the local cohomology can be described as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090111.png" /> be the set of prime ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090112.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090113.png" />. For an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090114.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090115.png" /> the submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090116.png" /> is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090117.png" />. Thus,
+
For an ideal $  \mathfrak a $
 +
in a commutative ring $  R $
 +
with unit element the local cohomology can be described as follows. Let $  A $
 +
be the set of prime ideals in $  R $
 +
containing $  \mathfrak a $.  
 +
For an $  R $-
 +
module $  M $
 +
the submodule $  \Gamma _ {A} ( M) $
 +
is defined as $  \{ {m } : {\textrm{ support } ( m) \subset  A } \} $.  
 +
Thus,
  
<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/l/l060/l060090/l060090118.png" /></td> </tr></table>
+
$$
 +
\Gamma _ {A} ( M)  = \{ {m } : { \mathop{\rm rad} (  \mathop{\rm Ann} ( m)) \supset \mathfrak a } \} =
 +
$$
  
<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/l/l060/l060090/l060090119.png" /></td> </tr></table>
+
$$
 +
= \
 +
\{ m : \mathfrak a  ^ {n} m = 0 \
 +
\textrm{ for }  n  \textrm{ large  enough  } \}  \simeq
 +
$$
  
<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/l/l060/l060090/l060090120.png" /></td> </tr></table>
+
$$
 +
\simeq \
 +
\lim\limits _ { {\  n \  } vec }  \mathop{\rm Hom} _ {R} ( R / \mathfrak a  ^ {n} , M ) .
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090121.png" /> is a covariant, left-exact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090122.png" />-linear functor from the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090123.png" />-modules into itself. Its derived functors are the local cohomology functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090124.png" /> (of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090125.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090126.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060090/l060090127.png" />)). These cohomology functors can be explicitly calculated using Koszul complexes, cf. [[Koszul complex|Koszul complex]].
+
$  M \mapsto \Gamma _ {A} ( M) $
 +
is a covariant, left-exact, $  R $-
 +
linear functor from the category of $  R $-
 +
modules into itself. Its derived functors are the local cohomology functors $  {\mathcal H} _ {A}  ^ {i} ( M) $(
 +
of $  M $
 +
with respect to $  A $(
 +
or $  \mathfrak a $)).  
 +
These cohomology functors can be explicitly calculated using Koszul complexes, cf. [[Koszul complex|Koszul complex]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> Y.-T. Siu,   "Techniques of extension of analytic objects" , M. Dekker (1974)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> Y.-T. Siu, "Techniques of extension of analytic objects" , M. Dekker (1974) {{MR|0361154}} {{ZBL|0294.32007}} </TD></TR></table>

Latest revision as of 22:17, 5 June 2020


with values in a sheaf of Abelian groups

A cohomology theory with values in a sheaf and with supports contained in a given subset. Let $ X $ be a topological space, $ {\mathcal F} $ a sheaf of Abelian groups on $ X $ and $ Z $ a locally closed subset of $ X $, that is, a closed subset of some subset $ V $ open in $ X $. Then $ \Gamma _ {Z} ( X , {\mathcal F} ) $ denotes the subgroup of $ \Gamma ( V , {\mathcal F} \mid _ {V} ) $ consisting of the sections of the sheaf $ {\mathcal F} \mid _ {V} $ with supports in $ Z $. If $ Z $ is fixed, then the correspondence $ {\mathcal F} \rightarrow \Gamma _ {Z} ( X , {\mathcal F} ) $ defines a left-exact functor from the category of sheaves of Abelian groups on $ X $ into the category of Abelian groups. The value of the corresponding $ i $- th right derived functor on $ {\mathcal F} $ is denoted by $ H _ {Z} ^ {i} ( X , {\mathcal F} ) $ and is called the $ i $- th local cohomology group of $ X $ with values in $ {\mathcal F} $, with respect to $ Z $. One has

$$ H _ {Z} ^ {0} ( X , {\mathcal F} ) = \Gamma _ {Z} ( X , {\mathcal F} ) . $$

Let $ {\mathcal H} _ {Z} ^ {0} ( {\mathcal F} ) $ be the sheaf on $ X $ corresponding to the pre-sheaf that associates with any open subset $ U \subset X $ the group $ \Gamma _ {Z \cap U } ( U , {\mathcal F} \mid _ {U} ) $. The correspondence $ {\mathcal F} \rightarrow {\mathcal H} _ {Z} ( {\mathcal F} ) $ is a left-exact functor from the category of sheaves of Abelian groups on $ X $ into itself. The value of its $ i $- th right derived functor on $ {\mathcal F} $ is denoted by $ {\mathcal H} _ {Z} ( {\mathcal F} ) $ and is called the $ i $- th local cohomology sheaf of $ {\mathcal F} $ with respect to $ Z $. The sheaf $ {\mathcal H} _ {Z} ^ {i} ( {\mathcal F} ) $ is associated with the pre-sheaf that associates with an open subset $ U \subset X $ the group $ H _ {Z \cap U } ^ {i} ( U , {\mathcal F} \mid _ {U} ) $.

There is a spectral sequence $ E _ {r} ^ {p,q} $, converging to $ H _ {Z} ^ {p+} q ( X , {\mathcal F} ) $, for which $ E _ {2} ^ {p,q} = H ^ {p} ( X , {\mathcal H} _ {Z} ^ {q} ( {\mathcal F} ) ) $( see [2], [3]).

Let $ Z $ be a locally closed subset of $ X $, $ Z ^ \prime $ a closed subset of $ Z $ and $ Z ^ {\prime\prime} = Z \setminus Z ^ \prime $; then there are the following exact sequences:

$$ \tag{1 } 0 \rightarrow H _ {Z ^ \prime } ^ {0} ( X , {\mathcal F} ) \rightarrow \dots \rightarrow \ H _ {Z ^ \prime } ^ {i} ( X , {\mathcal F} ) \rightarrow $$

$$ \rightarrow \ H _ {Z} ^ {i} ( X , {\mathcal F} ) \rightarrow H _ {Z ^ {\prime\prime} } ^ {i} ( X , {\mathcal F} ) \rightarrow H _ {Z ^ \prime } ^ {i+} 1 ( X , {\mathcal F} ) \rightarrow \dots ; $$

$$ \tag{2 } 0 \rightarrow {\mathcal H} _ {Z ^ \prime } ^ {0} ( {\mathcal F} ) \rightarrow \dots \rightarrow {\mathcal H} _ {Z ^ \prime } ^ {i} ( {\mathcal F} ) \rightarrow $$

$$ \rightarrow \ {\mathcal H} _ {Z} ^ {i} ( {\mathcal F} ) \rightarrow {\mathcal H} _ {Z ^ {\prime\prime} } ^ {i} ( {\mathcal F} ) \rightarrow {\mathcal H} _ {Z ^ \prime } ^ {i+} 1 ( {\mathcal F} ) \rightarrow \dots . $$

If $ Z $ is the whole of $ X $ and $ Z ^ \prime $ is a closed subset of $ X $, then the sequence (2) gives the exact sequence

$$ 0 \rightarrow {\mathcal H} _ {Z ^ \prime } ^ {0} ( {\mathcal F} ) \rightarrow {\mathcal F} \rightarrow \ {\mathcal H} _ {X \setminus Z ^ \prime } ^ {0} ( {\mathcal F} ) \rightarrow {\mathcal H} _ {Z ^ \prime } ^ {1} ( {\mathcal F} ) \rightarrow 0 $$

and the system of isomorphisms

$$ {\mathcal H} _ {X \setminus Z ^ \prime } ^ {i} ( {\mathcal F} ) \cong \ {\mathcal H} _ {Z ^ \prime } ^ {i+} 1 ( {\mathcal F} ) ,\ i \geq 1 . $$

The sheaves $ {\mathcal H} _ {X \setminus Z ^ \prime } ^ {i} ( {\mathcal F} ) $ are called the $ i $- th gap sheaves of $ {\mathcal F} $ and have important applications in questions concerning the extension of sections and cohomology classes of $ {\mathcal F} $, defined on $ X \setminus Z ^ \prime $, to the whole of $ X $( see [4]).

If $ X $ is a locally Noetherian scheme, $ {\mathcal F} $ is a quasi-coherent sheaf on $ X $ and $ Z $ is a closed subscheme of $ X $, then $ {\mathcal H} _ {Z} ^ {i} ( {\mathcal F} ) $ are quasi-coherent sheaves on $ X $. If $ {\mathcal Y} $ is a coherent sheaf of ideals on $ X $ that specifies the subscheme $ Z $, then one has the isomorphisms

$$ \lim\limits _ { {\ n \ } vec } \ \mathop{\rm Ext} _ { {\mathcal O} _ {X} } ^ {i} ( {\mathcal O} _ {X} / {\mathcal Y} ^ {n} , {\mathcal F} ) \cong \ {\mathcal H} _ {Z} ^ {i} ( {\mathcal F} ) . $$

The following criteria for triviality and coherence of local cohomology sheaves are important for applications (see [3], [4]).

Let $ X $ be a locally Noetherian scheme or a complex-analytic space, $ Z $ a locally closed subscheme or analytic subspace of $ X $, $ {\mathcal F} $ a coherent sheaf of $ {\mathcal O} _ {X} $- modules, and $ {\mathcal Y} $ a coherent sheaf of ideals that specifies $ Z $. Let

$$ \mathop{\rm prof} _ {Z} {\mathcal F} = \ \min _ {x \in Z } \ \mathop{\rm prof} _ { {\mathcal Y} _ {X,x} } {\mathcal F} _ {x} , $$

where $ \mathop{\rm prof} _ { {\mathcal Y} _ {X,x} } {\mathcal F} _ {x} $ is the maximal length of a sequence of elements of $ {\mathcal Y} _ {X,x} $ that is regular for $ {\mathcal F} _ {x} $, or $ \infty $ if $ {\mathcal F} _ {x} = 0 $. Then the equality $ {\mathcal H} _ {Z} ^ {i} ( {\mathcal F} ) = 0 $ for $ i < n $ is equivalent to the condition $ \mathop{\rm prof} _ {Z} {\mathcal F} \geq n $. Let $ \mathop{\rm codh} _ {x} {\mathcal F} _ {x} = \mathop{\rm prof} _ {\mathfrak m _ {x} } {\mathcal F} _ {x} $( where $ \mathfrak m $ is the maximal ideal of the ring $ {\mathcal O} _ {X,x} $) and let $ S _ {m} ( {\mathcal F} ) = \{ {x \in X } : { \mathop{\rm codh} _ {x} {\mathcal F} _ {x} \geq m } \} $. If $ X $ is a complex-analytic space or an algebraic variety, then all sets $ S _ {m} ( {\mathcal F} ) $ are analytic or algebraic, respectively. If $ {\mathcal F} $ is a coherent sheaf on $ X $ and $ Z $ is an analytic subspace or subvariety, respectively, then coherence of the sheaves $ {\mathcal H} _ {Z} ^ {i} ( {\mathcal F} ) $ for $ 0 \leq i \leq q $ is equivalent to the condition

$$ \mathop{\rm dim} Z \cap \overline{ {S _ {k+} q+ 1 }}\; ( {\mathcal F} \mid _ {X \setminus Z } ) \leq k $$

for any integer $ k $.

In terms of local cohomology one can define hyperfunctions, which have important applications in the theory of partial differential equations [5]. Let $ \Omega $ be an open subset of $ \mathbf R ^ {n} $, which is naturally imbedded in $ \mathbf C ^ {n} $. Then $ {\mathcal H} _ \Omega ^ {p} ( \mathbf C ^ {n} , {\mathcal O} _ {\mathbf C ^ {n} } ) = 0 $ for $ p \neq n $. The pre-sheaf $ \Omega \rightarrow {\mathcal H} _ \Omega ^ {n} ( \mathbf C ^ {n} , {\mathcal O} _ {\mathbf C ^ {n} } ) $ on $ \mathbf R ^ {n} $ defines a flabby sheaf, called the sheaf of hyperfunctions.

An analogue of local cohomology also exists in étale cohomology theory [3].

References

[1] I.V. Dolgachev, "Abstract algebraic geometry" Russian Math. Surveys , 2 : 3 (1974) pp. 264–303 Itogi Nauk. i Tekhn. Algebra. Topol. Geom. , 10 (1972) pp. 47–112 Zbl 1068.14059
[2] A. Grothendieck, "Local cohomology" , Lect. notes in math. , 41 , Springer (1967) MR0224620 Zbl 0185.49202
[3] A. Grothendieck, "Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux" , SGA 2 , North-Holland & Masson (1968) MR0476737 Zbl 1079.14001 Zbl 0159.50402
[4] Y.-T. Siu, "Gap-sheaves and extension of coherent analytic subsheaves" , Springer (1971) MR0287033 Zbl 0208.10403
[5] P. Schapira, "Théorie des hyperfonctions" , Lect. notes in math. , 126 , Springer (1970) MR0631543 MR0270151 Zbl 0201.44805 Zbl 0192.47305
[6] C. Banica, O. Stanasila, "Algebraic methods in the global theory of complex spaces" , Wiley (1976) (Translated from Rumanian) MR0463470 Zbl 0334.32001

Comments

See also Hyperfunction for the sheaf of hyperfunctions.

For an ideal $ \mathfrak a $ in a commutative ring $ R $ with unit element the local cohomology can be described as follows. Let $ A $ be the set of prime ideals in $ R $ containing $ \mathfrak a $. For an $ R $- module $ M $ the submodule $ \Gamma _ {A} ( M) $ is defined as $ \{ {m } : {\textrm{ support } ( m) \subset A } \} $. Thus,

$$ \Gamma _ {A} ( M) = \{ {m } : { \mathop{\rm rad} ( \mathop{\rm Ann} ( m)) \supset \mathfrak a } \} = $$

$$ = \ \{ m : \mathfrak a ^ {n} m = 0 \ \textrm{ for } n \textrm{ large enough } \} \simeq $$

$$ \simeq \ \lim\limits _ { {\ n \ } vec } \mathop{\rm Hom} _ {R} ( R / \mathfrak a ^ {n} , M ) . $$

$ M \mapsto \Gamma _ {A} ( M) $ is a covariant, left-exact, $ R $- linear functor from the category of $ R $- modules into itself. Its derived functors are the local cohomology functors $ {\mathcal H} _ {A} ^ {i} ( M) $( of $ M $ with respect to $ A $( or $ \mathfrak a $)). These cohomology functors can be explicitly calculated using Koszul complexes, cf. Koszul complex.

References

[a1] Y.-T. Siu, "Techniques of extension of analytic objects" , M. Dekker (1974) MR0361154 Zbl 0294.32007
How to Cite This Entry:
Local cohomology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_cohomology&oldid=11742
This article was adapted from an original article by D.A. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article