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

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