In the most general geometric form of the theory, sections \Gamma_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 \Gamma_Y form
local cohomology groups :H_Y^i(X,F) 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 \tilde M. 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 :\Gamma_I(M) := \bigcup_{n \ge 0} (0 :_M I^n), 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 H^i(\Gamma_I(E^\bullet)) of the
chain complex \Gamma_I(E^\bullet) obtained from taking the
I-torsion part \Gamma_I(-) of an
injective resolution E^\bullet of the module M. Because E^\bullet 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 \Gamma_I(M) may alternatively be described as :\Gamma_I(M) := \varinjlim_{n \in N} \operatorname {Hom}_R(R/I^n, M), and for this reason, the local cohomology of an
R-module
M agrees with a
direct limit of
Ext modules, :H_I^i(M) := \varinjlim_{n \in N} \operatorname {Ext}_R^i(R/I^n, M). It follows from either of these definitions that H^i_I(M) would be unchanged if I 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 M, 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 K^\bullet(f_1,\ldots,f_m) where f_1,\ldots, f_n generate I. The local cohomology modules can be described as: :H_I^i(M) \cong \varinjlim_m H^i \left (\operatorname{Hom}_R \left (K^\bullet \left (f_1^m, \dots, f_n^m \right ), M \right ) \right ) Koszul complexes have the property that multiplication by f_i induces a chain complex morphism \cdot f_i : K^\bullet(f_1,\ldots, f_n) \to K^\bullet(f_1,\ldots, f_n) that is homotopic to zero, meaning H^i(K^\bullet(f_1,\ldots, f_n)) is annihilated by the f_i. A non-zero map in the colimit of the \operatorname{Hom} 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 \check{C}^\bullet(f_1,\ldots,f_n;M), below. 0\to M \to \bigoplus_{i_0} M_{f_i} \to \bigoplus_{i_0 where the
ith local cohomology module of M with respect to I=(f_1,\ldots,f_n) is isomorphic to the
ith
cohomology group of the above
chain complex, :H^i_I(M)\cong H^i(\check{C}^\bullet(f_1,\ldots,f_n;M)). The broader issue of computing local cohomology modules (in
characteristic zero) is discussed in and . ==Basic properties==