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 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 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) MR0361154 Zbl 0294.32007