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.
axes in the affine plane, but cannot be extended to either 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
[1] Outside of algebraic geometry, local cohomology has found applications in commutative algebra,[2][3][4] combinatorics,[5][6][7] and certain kinds of partial differential equations.
[8] In the most general geometric form of the theory, sections
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
may alternatively be described as and for this reason, the local cohomology of an R-module M agrees[10] with a direct limit of Ext modules, It follows from either of these definitions that
[11] It also follows that local cohomology does not depend on any choice of generators for I, a fact which becomes relevant in the following definition involving the Čech complex.
The derived functor definition of local cohomology requires an injective resolution of the module
Iyengar et al. (2007), for example, state that they "essentially ignore" the "problem of actually producing any one of these [injective] kinds of resolutions for a given module"[12] prior to presenting the Čech complex definition of local cohomology, and Hartshorne (1977) describes Čech cohomology as "giv[ing] a practical method for computing cohomology of quasi-coherent sheaves on a scheme.
The local cohomology modules can be described[15] as: Koszul complexes have the property that multiplication by
sets contains maps from the all but finitely many Koszul complexes, and which are not annihilated by some element in the ideal.
is isomorphic to[18] the ith cohomology group of the above chain complex, The broader issue of computing local cohomology modules (in characteristic zero) is discussed in Leykin (2002) and Iyengar et al. (2007, Lecture 23).
Since local cohomology is defined as derived functor, for any short exact sequence of R-modules
For a quasicoherent sheaf F defined on X, this has the form In the setting where X is an affine scheme
and Y is the vanishing set of an ideal I, the cohomology groups
, this leads to an exact sequence where 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 Because of the above isomorphism with sheaf cohomology, local cohomology can be used to express a number of meaningful topological constructions on the scheme
For example, there is a natural analogue in local cohomology of the Mayer–Vietoris sequence with respect to a pair of open sets U and V in X, given by the complements of the closed subschemes corresponding to a pair of ideal I and J, respectively.
The vanishing of local cohomology can be used to bound the least number of equations (referred to as the arithmetic rank) needed to (set theoretically) define the algebraic set
[23] All of the basic properties of local cohomology expressed in this article are compatible with the graded structure.
is the ideal generated by all elements of positive degree (sometimes called the irrelevant ideal) is particularly special, due to its relationship with projective geometry.
Local cohomology can be used to prove certain upper bound results concerning the regularity.
The dimension dimR(M) of a module (defined as the Krull dimension of its support) provides an upper bound for local cohomology modules:[35] If R is local and M finitely generated, then this bound is sharp, i.e.,
The depth (defined as the maximal length of a regular M-sequence; also referred to as the grade of M) provides a sharp lower bound, i.e., it is the smallest integer n such that[36] These two bounds together yield a characterisation of Cohen–Macaulay modules over local rings: they are precisely those modules where
that is a homomorphic image of a Gorenstein local ring[37] (for example, if
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.
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 (i.e., the minimal dimension of a closed subset T of Z that has to be removed from Z so that the complement Z \ T is disconnected) is bound by For example, Z is connected if dim V + dim W > r.[42] In polyhedral geometry, a key ingredient of Stanley’s 1975 proof of the simplicial form of McMullen’s Upper bound theorem involves showing that the Stanley-Reisner ring of the corresponding simplicial complex is Cohen-Macaulay, and local cohomology is an important tool in this computation, via Hochster’s formula.