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Concept
Local cohomology
Summary
In algebraic geometry, local cohomology is an algebraic analogue 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 . Given a function (more generally, a section of a quasicoherent sheaf) defined on an open subset of an algebraic variety (or scheme), local cohomology measures the obstruction to extending that function to a larger domain. The rational function , for example, is defined only on the complement of on the affine line over a field , and cannot be extended to a function on the entire space. The local cohomology module (where is the coordinate ring of ) detects this in the nonvanishing of a cohomology class . In a similar manner, is defined away from the and axes in the affine plane, but cannot be extended to either the complement of the -axis or the complement of the -axis alone (nor can it be expressed as a sum of such functions); this obstruction corresponds precisely to a nonzero class in the local cohomology module .
Outside of algebraic geometry, local cohomology has found applications in commutative algebra, combinatorics, and certain kinds of partial differential equations.
In the most general 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
In the theory's algebraic form, the space X is the spectrum Spec(R) of a commutative ring R (assumed to be Noetherian throughout this article) 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(F) corresponds to the I-torsion functor, a union of annihilators
i.e., the elements of M which are annihilated by some power of I. As a right derived functor, the ith local cohomology module with respect to I is the ith cohomology group of the chain complex obtained from taking the I-torsion part of an injective resolution of the module .
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