Local cohomology

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 formula_1, for example, is defined only on the complement of formula_2 on the affine line formula_3 over a field formula_4, and cannot be extended to a function on the entire space. The local cohomology module formula_5 (where formula_6 is the coordinate ring of formula_3) detects this in the nonvanishing of a cohomology class formula_8. In a similar manner, formula_9 is defined away from the formula_10 and formula_11 axes in the affine plane, but cannot be extended to either the complement of the formula_10-axis or the complement of the formula_11-axis alone (nor can it be expressed as a sum of such functions); this obstruction corresponds precisely to a nonzero class formula_14 in the local cohomology module formula_15.
Outside of algebraic geometry, local cohomology has found applications in commutative algebra, combinatorics
, and certain kinds of partial differential equations.
Definition.
In the most general geometric form of the theory, sections formula_16 are considered of a sheaf formula_17 of abelian groups, on a topological space formula_18, with support in a closed subset formula_19, The derived functors of formula_16 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 formula_22. The closed subscheme "Y" is defined by an ideal "I". In this situation, the functor Γ"Y"("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 "i"th local cohomology module with respect to "I" is the "i"th cohomology group formula_24 of the chain complex formula_25 obtained from taking the "I"-torsion part formula_26 of an injective resolution formula_27 of the module formula_28. Because formula_27 consists of "R"-modules and "R"-module homomorphisms, the local cohomology groups each have the natural structure of an "R"-module.
The "I"-torsion part formula_30 may alternatively be described as
and for this reason, the local cohomology of an "R"-module "M" agrees with a direct limit of "Ext" modules,
It follows from either of these definitions that formula_33 would be unchanged if formula_34 were replaced by another ideal having the same radical. 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.
Using Koszul and Čech complexes.
The derived functor definition of local cohomology requires an injective resolution of the module formula_28, which can make it inaccessible for use in explicit computations. The Čech complex is seen as more practical in certain contexts. , for example, state that they "essentially ignore" the "problem of actually producing any one of these [injective] kinds of resolutions for a given module" prior to presenting the Čech complex definition of local cohomology, and describes Čech cohomology as "giv[ing] a practical method for computing cohomology of quasi-coherent sheaves on a scheme." and as being "well suited for computations."
The Čech complex can be defined as a colimit of Koszul complexes formula_36 where formula_37 generate formula_34. The local cohomology modules can be described as:
Koszul complexes have the property that multiplication by formula_40 induces a chain complex morphism formula_41 that is homotopic to zero, meaning formula_42 is annihilated by the formula_40. A non-zero map in the colimit of the formula_44 sets contains maps from the all but finitely many Koszul complexes, and which are not annihilated by some element in the ideal. 
This colimit of Koszul complexes is isomorphic to the Čech complex, denoted formula_45, below. formula_46 
where the "i"th local cohomology module of formula_28 with respect to formula_48 is isomorphic to the "i"th cohomology group of the above chain complex,
The broader issue of computing local cohomology modules (in characteristic zero) is discussed in and .
Basic properties.
Since local cohomology is defined as derived functor, for any short exact sequence of "R"-modules formula_50, there is, by definition, a natural long exact sequence in local cohomology
There is also 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 modules. For a quasicoherent sheaf "F" defined on "X", this has the form
In the setting where "X" is an affine scheme formula_53 and "Y" is the vanishing set of an ideal "I", the cohomology groups formula_54 vanish for formula_55. If formula_56, 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 formula_59 in purely algebraic terms. 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. This sequence has the form
for any formula_61-module formula_28.
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 formula_63 in formula_64. If formula_65 has the same radical as formula_34, and is generated by formula_67 elements, then the Čech complex on the generators of formula_65 has no terms in degree formula_69. The least number of generators among all ideals formula_65 such that formula_71 is the arithmetic rank of formula_34, denoted formula_73. Since the local cohomology with respect to formula_34 may be computed using any such ideal, it follows that formula_75 for formula_76.
Graded local cohomology and projective geometry.
When formula_61 is graded by formula_78, formula_34 is generated by homogeneous elements, and formula_28 is a graded module, there is a natural grading on the local cohomology module formula_33 that is compatible with the gradings of formula_28 and formula_61. All of the basic properties of local cohomology expressed in this article are compatible with the graded structure. If formula_28 is finitely generated and formula_85 is the ideal generated by the elements of formula_61 having positive degree, then the graded components formula_87 are finitely generated over formula_61 and vanish for sufficiently large formula_67.
The case where formula_90 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. In this case, there is an isomorphism
where formula_92 is the projective scheme associated to formula_61, and formula_94 denotes the Serre twist. This isomorphism is graded, giving
in all degrees formula_67.
This isomorphism relates local cohomology with the global cohomology of projective schemes. For example, the Castelnuovo–Mumford regularity can be formulated using local cohomology as
where formula_98 denotes the highest degree formula_99 such that formula_100. Local cohomology can be used to prove certain upper bound results concerning the regularity.
Examples.
Top local cohomology.
Using the Čech complex, if formula_101 the local cohomology module formula_102 is generated over formula_61 by the images of the formal fractions
for formula_105 and formula_106. This fraction corresponds to a nonzero element of formula_102 if and only if there is no formula_108 such that formula_109. For example, if formula_110, then
Examples of H1.
If formula_127 is known (where formula_128), the module formula_129 can sometimes be computed explicitly using the sequence
In the following examples, formula_4 is any field.
Relation to invariants of modules.
The dimension dim"R"(M) of a module (defined as the Krull dimension of its support) provides an upper bound for local cohomology modules:
If "R" is local and "M" finitely generated, then this bound is sharp, i.e., formula_148.
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
These two bounds together yield a characterisation of Cohen–Macaulay modules over local rings: they are precisely those modules where formula_150 vanishes for all but one "n".
Local duality.
The local duality theorem is a local analogue of Serre duality. For a Cohen-Macaulay local ring formula_61 of dimension formula_152 that is a homomorphic image of a Gorenstein local ring (for example, if formula_61 is complete), it states that the natural pairing
is a perfect pairing, where formula_155 is a dualizing module for formula_61. In terms of the Matlis duality functor formula_157, the local duality theorem may be expressed as the following isomorphism.
The statement is simpler when formula_159, which is equivalent to the hypothesis that formula_61 is Gorenstein. This is the case, for example, if formula_61 is regular.
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 (a local analogue of the Bertini theorem) or the Fulton–Hansen connectedness theorem due to and . The latter asserts that for two projective varieties "V" and "W" in P"r" 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".
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.