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 Hartshorne (1967), and in 1961-2 at IHES written up as SGA2 - Grothendieck (1968), republished as Grothendieck (2005).

In the geometric form of the theory, sections Γ_Y 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 Γ_Y form local cohomology groups

H_Yⁱ(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.

References

• M. P. Brodman and R. Y. Sharp (1998) Local Cohomology: An Algebraic Introduction with Geometric Applications Book review by Hartshorne
• Grothendieck, Alexander (2005) [1968], Séminaire de Géométrie Algébrique du Bois Marie - 1962 - Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux - (SGA 2), Documents Mathématiques (Paris), 4, Paris: Société Mathématique de France, arXiv:math/0511279, ISBN 978-2-85629-169-6, MR 2171939 
• Grothendieck, Alexandre (1968) [1962]. Séminaire de Géométrie Algébrique du Bois Marie - 1962 - Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux - (SGA 2) (Advanced Studies in Pure Mathematics 2) (in French). Amsterdam: North-Holland Publishing Company. vii+287. 
• Hartshorne, Robin (1967) [1961], Local cohomology. A seminar given by A. Grothendieck, Harvard University, Fall, 1961, Lecture notes in mathematics, 41, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0073971, MR 0224620 
• Iyengar, Srikanth B.; Leuschke, Graham J.; Leykin, Anton; Miller, Claudia; Miller, Ezra; Singh, Anurag K.; Walther, Uli (2007), Twenty-four hours of local cohomology, Graduate Studies in Mathematics, 87, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4126-6, MR 2355715