In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, Čech cohomology is a
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be view ...
theory based on the intersection properties of
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* Open (Blues Image album), ''Open'' (Blues Image album), 1969
* Open (Gotthard album), ''Open'' (Gotthard album), 1999
* Open (C ...
covers of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
. It is named for the mathematician
Eduard Čech.
Motivation
Let ''X'' be a topological space, and let
be an open cover of ''X''. Let
denote the
nerve
A nerve is an enclosed, cable-like bundle of nerve fibers (called axons) in the peripheral nervous system.
A nerve transmits electrical impulses. It is the basic unit of the peripheral nervous system. A nerve provides a common pathway for the ...
of the covering. The idea of Čech cohomology is that, for an open cover
consisting of sufficiently small open sets, the resulting simplicial complex
should be a good combinatorial model for the space ''X''. For such a cover, the Čech cohomology of ''X'' is defined to be the
simplicial cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be view ...
of the nerve. This idea can be formalized by the notion of a
good cover. However, a more general approach is to take the
direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cate ...
of the cohomology groups of the nerve over the system of all possible open covers of ''X'', ordered by
refinement. This is the approach adopted below.
Construction
Let ''X'' be a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
, and let
be a
presheaf of
abelian group
In mathematics, 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 the order in which they are written. That is, the group operation is comm ...
s on ''X''. Let
be an
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alp ...
of ''X''.
Simplex
A ''q''-simplex σ of
is an ordered collection of ''q''+1 sets chosen from
, such that the intersection of all these sets is non-empty. This intersection is called the ''support'' of σ and is denoted , σ, .
Now let
be such a ''q''-simplex. The ''j-th partial boundary'' of σ is defined to be the (''q''−1)-simplex obtained by removing the ''j''-th set from σ, that is:
:
The ''boundary'' of σ is defined as the alternating sum of the partial boundaries:
:
viewed as an element of the
free abelian group
In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...
spanned by the simplices of
.
Cochain
A ''q''-cochain of
with coefficients in
is a map which associates with each ''q''-simplex σ an element of
and we denote the set of all ''q''-cochains of
with coefficients in
by
.
is an abelian group by pointwise addition.
Differential
The cochain groups can be made into a
cochain 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 th ...
by defining the coboundary operator
by:
where
is the
restriction morphism from
to
(Notice that ∂
jσ ⊆ σ, but σ ⊆ ∂
jσ.)
A calculation shows that
The
coboundary operator is analogous to the
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
of
De Rham cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adap ...
, so it sometimes called
the differential of the
cochain 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 th ...
.
Cocycle
A ''q''-cochain is called a ''q''-cocycle if it is in the kernel of
, hence
is the set of all ''q''-cocycles.
Thus a (''q''−1)-cochain
is a cocycle if for all ''q''-simplices
the cocycle condition
:
holds.
A 0-cocycle
is a collection of local sections of
satisfying a compatibility relation on every intersecting
:
A 1-cocycle
satisfies for every non-empty
with
:
Coboundary
A ''q''-cochain is called a ''q''-coboundary if it is in the image of
and
is the set of all ''q''-coboundaries.
For example, a 1-cochain
is a 1-coboundary if there exists a 0-cochain
such that for every intersecting
:
Cohomology
The Čech cohomology of
with values in
is defined to be the cohomology of the cochain complex
. Thus the ''q''th Čech cohomology is given by
:
.
The Čech cohomology of ''X'' is defined by considering
refinements of open covers. If
is a refinement of
then there is a map in cohomology
The open covers of ''X'' form a
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
under refinement, so the above map leads to a
direct system of abelian groups. The Čech cohomology of ''X'' with values in ''
'' is defined as the
direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cate ...
of this system.
The Čech cohomology of ''X'' with coefficients in a fixed abelian group ''A'', denoted
, is defined as
where
is the
constant sheaf
Constant or The Constant may refer to:
Mathematics
* Constant (mathematics), a non-varying value
* Mathematical constant, a special number that arises naturally in mathematics, such as or
Other concepts
* Control variable or scientific const ...
on ''X'' determined by ''A''.
A variant of Čech cohomology, called numerable Čech cohomology, is defined as above, except that all open covers considered are required to be ''numerable'': that is, there is a
partition of unity
In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X:
* there is a neighbourhood of where all but a finite number of the functions of are 0 ...
such that each support
is contained in some element of the cover. If ''X'' is
paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
and
Hausdorff, then numerable Čech cohomology agrees with the usual Čech cohomology.
Relation to other cohomology theories
If ''X'' is
homotopy equivalent
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
to a
CW complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
, then the Čech cohomology
is
naturally isomorphic
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
to the
singular cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
. If ''X'' is a
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, then
is also naturally isomorphic to the
de Rham cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adap ...
; the article on de Rham cohomology provides a brief review of this isomorphism. For less well-behaved spaces, Čech cohomology differs from singular cohomology. For example if ''X'' is the
closed topologist's sine curve, then
whereas
If ''X'' is a differentiable manifold and the cover
of ''X'' is a "good cover" (''i.e.'' all the sets ''U''
α are
contractible
In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...
to a point, and all finite intersections of sets in
are either empty or contractible to a point), then
is isomorphic to the de Rham cohomology.
If ''X'' is compact Hausdorff, then Čech cohomology (with coefficients in a discrete group) is isomorphic to
Alexander-Spanier cohomology.
In algebraic geometry
Čech cohomology can be defined more generally for objects in a
site
Site most often refers to:
* Archaeological site
* Campsite, a place used for overnight stay in an outdoor area
* Construction site
* Location, a point or an area on the Earth's surface or elsewhere
* Website, a set of related web pages, typical ...
C endowed with a topology. This applies, for example, to the Zariski site or the etale site of a
scheme A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
''X''. The Čech cohomology with values in some
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* ''The Sheaf'', a student-run newspaper se ...
''F'' is defined as
:
where the
colimit
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such ...
runs over all coverings (with respect to the chosen topology) of ''X''. Here
is defined as above, except that the ''r''-fold intersections of open subsets inside the ambient topological space are replaced by the ''r''-fold
fiber product
In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is often w ...
:
As in the classical situation of topological spaces, there is always a map
:
from Čech cohomology to
sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally whe ...
. It is always an isomorphism in degrees ''n'' = 0 and 1, but may fail to be so in general. For the
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
on a
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
separated scheme, Čech and sheaf cohomology agree for any
quasi-coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
. For the
étale topology In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic. The étale to ...
, the two cohomologies agree for any étale sheaf on ''X'', provided that any finite set of points of ''X'' are contained in some open affine subscheme. This is satisfied, for example, if ''X'' is
quasi-projective In mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A similar definition is used in ...
over an
affine scheme
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...
.
The possible difference between Cech cohomology and sheaf cohomology is a motivation for the use of
hypercovering In mathematics, and in particular homotopy theory, a hypercovering (or hypercover) is a simplicial object that generalises the Čech nerve of a cover. For the Čech nerve of an open cover one can show that if the space X is compact and if every ...
s: these are more general objects than the Cech
nerve
A nerve is an enclosed, cable-like bundle of nerve fibers (called axons) in the peripheral nervous system.
A nerve transmits electrical impulses. It is the basic unit of the peripheral nervous system. A nerve provides a common pathway for the ...
:
A hypercovering ''K''
∗ of ''X'' is a
simplicial object in C, i.e., a collection of objects ''K''
''n'' together with boundary and degeneracy maps. Applying a sheaf ''F'' to ''K''
∗ yields a
simplicial abelian group In mathematics, more precisely, in the theory of simplicial sets, a simplicial group is a simplicial object in the category of groups. Similarly, a simplicial abelian group is a simplicial object in the category of abelian groups. A simplicial grou ...
''F''(''K''
∗) whose ''n''-th cohomology group is denoted ''H''
''n''(''F''(''K''
∗)). (This group is the same as
in case ''K'' equals
.) Then, it can be shown that there is a canonical isomorphism
:
where the colimit now runs over all hypercoverings.
[, Theorem 8.16]
Examples
For example, we can compute the coherent sheaf cohomology of
on the projective line
using the Čech complex. Using the cover
:
we have the following modules from the cotangent sheaf
:
If we take the conventions that
then we get the Čech complex
:
Since
is injective and the only element not in the image of
is
we get that
:
References
Citation footnotes
General references
*
*
* . . Chapter 2 Appendix A
{{DEFAULTSORT:Cech cohomology
Cohomology theories