Local cohomology
Encyclopedia
In mathematics
, local cohomology is a chapter of homological algebra
and sheaf theory introduced into algebraic geometry
by Alexander Grothendieck
. He developed it in seminars in 1961 at Harvard University
, and 1961-2 at IHES. It was later written up as SGA2. Applications to commutative algebra
and hyperfunction
theory followed.
In the geometric form of the theory, sections ΓY are considered of a sheaf
F of abelian group
s, on a topological space
X, with support
in a closed subset Y. The derived functor
s of ΓY form local cohomology groups
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 theorem
s. 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 ring
s 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, local cohomology is a chapter of homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
and sheaf theory introduced into algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
by Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
. He developed it in seminars in 1961 at Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
, and 1961-2 at IHES. It was later written up as SGA2. Applications to commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
and hyperfunction
Hyperfunction
In mathematics, hyperfunctions are generalizations of functions, as a 'jump' from one holomorphic function to another at a boundary, and can be thought of informally as distributions of infinite order...
theory followed.
In the geometric form of the theory, sections ΓY are considered of a sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
F of abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
s, on a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X, with support
Support (mathematics)
In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...
in a closed subset Y. The derived functor
Derived functor
In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.- Motivation :...
s of ΓY form local cohomology groups
- HYi(X,F)
There is a long exact sequence of sheaf cohomology
Sheaf cohomology
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
linking the ordinary sheaf cohomology of X and of the open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
U = X \Y, with the local cohomology groups.
The initial applications were to analogues of the Lefschetz hyperplane theorem
Lefschetz hyperplane theorem
In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties...
s. In general such theorems state that homology or cohomology is supported on a hyperplane section
Hyperplane section
In mathematics, a hyperplane section of a subset X of projective space Pn is the intersection of X with some hyperplane H — in other words we look at the subset XH of those elements x of X that satisfy the single linear condition L = 0 defining H as a linear subspace...
of an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
, except for some 'loss' that can be controlled. These results applied to the algebraic fundamental group and to the Picard group.
In commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
for a commutative ring R and its spectrum
Spectrum of a ring
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
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
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
M, which gives a quasicoherent sheaf on Spec(R). In this setting the depth of a module can be characterised over local ring
Local ring
In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
s by the vanishing of local cohomology groups, and there is an analogue, the local duality theorem, of Serre duality
Serre duality
In algebraic geometry, a branch of mathematics, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n . It shows that a cohomology group Hi is the dual space of another one, Hn−i...
, using Ext functors of R-modules and a dualising module.