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''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  $  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

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

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 [[#References|[2]]], [[#References|[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 [[#References|[4]]]).

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

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

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

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 [[#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]]].

====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 {{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====
See also [[Hyperfunction|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|Koszul complex]].

====References====
<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>