In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support.
Singular cohomology with compact support
Let
be a topological space. Then
:
This is also naturally isomorphic to the cohomology of the sub–
chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of t ...
consisting of all singular
cochain
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of th ...
s
that have compact support in the sense that there exists some compact
such that
vanishes on all chains in
.
Functorial definition
Let
be a topological space and
the map to the point. Using the
direct image In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf ''F'' defined on a topolo ...
and
direct image with compact support In mathematics, the direct image with compact (or proper) support is an image functor for sheaves that extends the compactly supported global sections functor to the relative setting. It is one of Grothendieck's six operations.
Definition
Le ...
functors
, one can define cohomology and cohomology with compact support of a sheaf of abelian groups
on
as
:
:
Taking for
the constant sheaf with coefficients in a ring
recovers the previous definition.
de Rham cohomology with compact support for smooth manifolds
Given a manifold ''X'', let
be the
real vector space
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
of ''k''-forms on ''X'' with compact support, and ''d'' be the standard
exterior derivative. Then the de Rham cohomology groups with compact support
are the
homology of the
chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of t ...
:
:
''i.e.'',
is the vector space of
closed ''q''-forms
modulo that of exact ''q''-forms.
Despite their definition as the homology of an ascending complex, the de Rham groups with compact support demonstrate
covariant behavior; for example, given the inclusion mapping ''j'' for an open set ''U'' of ''X'', extension of forms on ''U'' to ''X'' (by defining them to be 0 on ''X''–''U'') is a map
inducing a map
:
.
They also demonstrate contravariant behavior with respect to
proper map
In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.
Definition
There are several competing definit ...
s - that is, maps such that the inverse image of every compact set is compact. Let ''f'': ''Y'' → ''X'' be such a map; then the
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: i ...
:
induces a map
:
.
If ''Z'' is a submanifold of ''X'' and ''U'' = ''X''–''Z'' is the complementary open set, there is a long exact sequence
:
called the long exact sequence of cohomology with compact support. It has numerous applications, such as the
Jordan curve theorem
In topology, the Jordan curve theorem asserts that every '' Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an " exterior" region containing all of the nearby and far away exteri ...
, which is obtained for ''X'' = R² and ''Z'' a simple closed curve in ''X''.
De Rham cohomology with compact support satisfies a covariant
Mayer–Vietoris sequence
In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is due ...
: if ''U'' and ''V'' are open sets covering ''X'', then
:
where all maps are induced by extension by zero is also exact.
See also
*
Borel–Moore homology
*
Poincaré duality
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...
*
Constructible sheaf In mathematics, a constructible sheaf is a sheaf of abelian groups over some topological space ''X'', such that ''X'' is the union of a finite number of locally closed subsets on each of which the sheaf is a locally constant sheaf. It has its origi ...
*
Derived category
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction pr ...
References
*
*
*{{cite web , title=Cohomology with support and Poincare duality , url=https://math.stackexchange.com/q/2732445 , website=Stack Exchange
Cohomology theories