# Local cohomology

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

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=47679
This article was adapted from an original article by D.A. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article