# 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 be a topological space, a sheaf of Abelian groups on and a locally closed subset of , that is, a closed subset of some subset open in . Then denotes the subgroup of consisting of the sections of the sheaf with supports in . If is fixed, then the correspondence defines a left-exact functor from the category of sheaves of Abelian groups on into the category of Abelian groups. The value of the corresponding -th right derived functor on is denoted by and is called the -th local cohomology group of with values in , with respect to . One has

Let be the sheaf on corresponding to the pre-sheaf that associates with any open subset the group . The correspondence is a left-exact functor from the category of sheaves of Abelian groups on into itself. The value of its -th right derived functor on is denoted by and is called the -th local cohomology sheaf of with respect to . The sheaf is associated with the pre-sheaf that associates with an open subset the group .

There is a spectral sequence , converging to , for which (see [2], [3]).

Let be a locally closed subset of , a closed subset of and ; then there are the following exact sequences:

(1) |

(2) |

If is the whole of and is a closed subset of , then the sequence (2) gives the exact sequence

and the system of isomorphisms

The sheaves are called the -th gap sheaves of and have important applications in questions concerning the extension of sections and cohomology classes of , defined on , to the whole of (see [4]).

If is a locally Noetherian scheme, is a quasi-coherent sheaf on and is a closed subscheme of , then are quasi-coherent sheaves on . If is a coherent sheaf of ideals on that specifies the subscheme , then one has the isomorphisms

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

Let be a locally Noetherian scheme or a complex-analytic space, a locally closed subscheme or analytic subspace of , a coherent sheaf of -modules, and a coherent sheaf of ideals that specifies . Let

where is the maximal length of a sequence of elements of that is regular for , or if . Then the equality for is equivalent to the condition . Let (where is the maximal ideal of the ring ) and let . If is a complex-analytic space or an algebraic variety, then all sets are analytic or algebraic, respectively. If is a coherent sheaf on and is an analytic subspace or subvariety, respectively, then coherence of the sheaves for is equivalent to the condition

for any integer .

In terms of local cohomology one can define hyperfunctions, which have important applications in the theory of partial differential equations [5]. Let be an open subset of , which is naturally imbedded in . Then for . The pre-sheaf on 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 |

[2] | A. Grothendieck, "Local cohomology" , Lect. notes in math. , 41 , Springer (1967) |

[3] | A. Grothendieck, "Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux" , SGA 2 , North-Holland & Masson (1968) |

[4] | Y.-T. Siu, "Gap-sheaves and extension of coherent analytic subsheaves" , Springer (1971) |

[5] | P. Schapira, "Théorie des hyperfonctions" , Lect. notes in math. , 126 , Springer (1970) |

[6] | C. Banica, O. Stanasila, "Algebraic methods in the global theory of complex spaces" , Wiley (1976) (Translated from Rumanian) |

#### Comments

See also Hyperfunction for the sheaf of hyperfunctions.

For an ideal in a commutative ring with unit element the local cohomology can be described as follows. Let be the set of prime ideals in containing . For an -module the submodule is defined as . Thus,

is a covariant, left-exact, -linear functor from the category of -modules into itself. Its derived functors are the local cohomology functors (of with respect to (or )). 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) |

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Local cohomology.

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