Local cohomology

In algebraic geometry, local cohomology is an analog of relative cohomology. Alexander Grothendieck introduced it in seminars in 1961 written up by Hartshorne (1967), and in 1961-2, later written up as SGA2 by Grothendieck & Raynaud (2005).

In the geometric form of the theory, sections ΓY are considered of a sheaf F of abelian groups, on a topological space X, with support in a closed subset Y. The derived functors of ΓY form local cohomology groups

HYi(X,F)

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.

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.

In commutative algebra for a commutative ring R and its spectrum Spec(R) as X, Y can be replaced by the closed subscheme defined by an ideal I of R. The sheaf F can be replaced by an R-module M, which gives a quasicoherent sheaf on Spec(R). In this setting the depth of a module can be characterised over local rings by the vanishing of local cohomology groups, and there is an analogue, the local duality theorem, of Serre duality, using Ext functors of R-modules and a dualising module.

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