Local cohomology


In algebraic geometry, local cohomology is an analog of relative cohomology. Alexander Grothendieck introduced it in seminars in Harvard in 1961 written up by, and in 1961-2 at IHES written up as SGA2 -, republished as.

Definition

In the geometric form of the theory, sections are considered of a sheaf of abelian groups, on a topological space, with support in a closed subset, The derived functors of form local cohomology groups
For applications in commutative algebra, the space X is the spectrum Spec of a commutative ring R and the sheaf F is the quasicoherent sheaf associated to an R-module M, denoted by. The closed subscheme Y is defined by an ideal I. In this situation, the functor ΓY corresponds to the annihilator
i.e., the elements of M which are annihilated by some power of I. Equivalently,
which also shows that local cohomology of quasi-coherent sheaves agrees with

Using Koszul complexes

For an ideal, the local cohomology groups can be computed by means of a colimit of Koszul complexes:
Because Koszul complexes have the property that multiplication by as a chain complex morphism is homotopic to zero, meaning is annihilated by the, a non-zero map in the colimit of the hom sets contain maps from the all but finitely many Koszul complexes, and which are not annihilated by some element in the ideal.
Also, this colimit of Koszul complexes can be computed to be the Cech complex

Basic properties

There is a long exact sequence of sheaf cohomology linking the ordinary sheaf cohomology of X and of the open set U = X \Y, with the local cohomology groups.
In particular, this leads to an exact sequence
where U is the open complement of Y and the middle map is the restriction of sections. The target of this restriction map is also referred to as the ideal transform. For n ≥ 1, there are isomorphisms
An important special case is the one when R is graded, I consists of the elements of degree ≥ 1, and M is a graded module. In this case, the cohomology of U above can be identified with the cohomology groups
of the projective scheme associated to R and denotes the Serre twist. This relates local cohomology with global cohomology on projective schemes. For example, Castelnuovo–Mumford regularity can be formulated using local cohomology.

Relation to invariants of modules

The dimension dimR of a module provides an upper bound for local cohomology groups:
If R is local and M finitely generated, then this bound is sharp, i.e.,.
The depth provides a sharp lower bound, i.e., it is the smallest integer n such that
These two bounds together yield a characterisation of Cohen–Macaulay modules over local rings: they are precisely those modules where vanishes for all but one n.

Local duality

The local duality theorem is a local analogue of Serre duality. For a complete Cohen-Macaulay local ring R, it states that the natural pairing
is a perfect pairing, where is a dualizing module for R.

Applications

The initial applications were to analogues of the Lefschetz hyperplane theorems. In general such theorems state that homology or cohomology is supported on a hyperplane section of an algebraic variety, except for some 'loss' that can be controlled. These results applied to the algebraic fundamental group and to the Picard group.
Another type of application are connectedness theorems such as Grothendieck's connectedness theorem or the Fulton–Hansen connectedness theorem due to and. The latter asserts that for two projective varieties V and W in Pr over an algebraically closed field, the connectedness dimension of Z = V ∩ W is bound by
For example, Z is connected if dim V + dim W > r.