A special mathematical tool which provides a unified approach
to establishing connections between local and global properties of
topological spaces (in particular geometric objects) and which is a
powerful method for studying many problems in
contemporary algebra, geometry, topology, and analysis.
A
pre-sheaf
on a topological space
assigns to each open subset
an Abelian group
(a ring, a module over a ring, etc.) and to every pair of open sets
a homomorphism
,
such that
is the identity isomorphism and
for every triple
.
In other words, a pre-sheaf is a contravariant
functor from the category of open subsets of
and their inclusions into the category of groups
(rings, etc.) and their homomorphisms. The mappings
are called
restriction homomorphisms
(for example, if the elements of the
stalk
are functions of some type or other defined on
,
is the restriction of these to the smaller subset). A topology on the set
,
where
is, by definition, the direct limit
,
is defined in the following way: For each
and any
,
the set
consisting of those points of
,
,
which are images of
in the definition of
is declared to be open in
.
In this topology the stalks
are discrete, the stalk-wise algebraic operations defined on
by taking direct limits are continuous and the natural projection
,
where
,
is a local homeomorphism. The space
together with the stalk-wise algebraic operations and the projection
is called the
sheaf
of Abelian groups (rings, etc.) over
associated with the pre-sheaf
.
Every continuous mapping
for which
is called a
section
of
over
.
The section of
over
defined by the zeros in
is called the
zero section.
If a section
is zero at a point
,
then
coincides with the zero section in some neighbourhood of
;
therefore the set of points at which
is not zero (the
support
of
)
is closed in
.
Let
(respectively,
,
where
is a certain family of closed sets in
;
in particular,
)
be the group (ring, module, etc.) of all sections of
over
(respectively, all sections over
with supports in
;
in particular, sections with compact support). The assignment
is a pre-sheaf over
,
called the
pre-sheaf of sections
of the sheaf
.
The assignment
used in defining the topology on
also defines homomorphisms
which commute with the restrictions to
,
that is, it defines a homomorphism of pre-sheaves. This
homomorphism is an isomorphism provided that the original pre-sheaf
satisfies the requirements: a) if
and
,
then
if the restrictions of
and
to each
are equal; and b) if
and
is a collection of elements such that the restrictions of
and
to
coincide, then there exists a
which has restriction to each
coinciding with
.
The concept of a pre-sheaf satisfying these requirements
is equivalent to the concept of the sheaf associated
with it, therefore such pre-sheaves are quite often called
sheaves
also.
A sheaf of the form
(with the evident projection to
),
where
is a fixed group (ring, etc.), is called a
constant sheaf,
and is denoted by
.
A sheaf which is constant in sufficiently small neighbourhoods of every
is called
locally constant.
The topology of such sheaves is separated (i.e. Hausdorff) if
is a separated space. In more typical situations the topology of
can be non-separated even if
is separated (such is the case, for example, for the sheaf of
germs of continuous (or differentiable) functions which is generated by the pre-sheaf
,
where
is the set of continuous (differentiable) functions on
;
however, the sheaf of germs of analytic functions on a manifold is separated).
Every homomorphism of pre-sheaves
induces a mapping of the associated sheaves
,
which is a local homeomorphism and maps stalks homomorphically to
stalks; such a mapping of sheaves is called a
sheaf homomorphism.
Mono- and epimorphisms are defined in the standard way. For any sheaf homomorphism
the image
is an open subset of
,
closed with respect to the stalk-wise algebraic operations. Every subset of
which satisfies these requirements is called a
subsheaf
of
.
The
quotient sheaf
of the sheaf
by a subsheaf
is defined as the sheaf
associated with the pre-sheaf
;
moreover, there is an epimorphism
,
and
.
For every open
there is a subsheaf in
,
denoted by
,
which is the union of
with the zero section of
over
;
denotes the corresponding quotient sheaf (whose restriction to
coincides with the restriction thereto of
).
Since it is possible to interpret such ordinary terms as
homomorphism, kernel, image, subsheaf, quotient sheaf, etc. for sheaves over
in such a way that these concepts have essentially the same
meaning as in algebra, one can consider them from a categorical
point of view and apply to sheaf theory the constructions of
homological algebra.
The resulting category of sheaves over
has the same classical properties as the category of Abelian
groups or the category of modules; in particular, one can
define for sheaves direct sums, infinite direct
products, inductive limits, and other concepts.
The apparatus of sheaf theory has penetrated into various fields of mathematics
thanks to the fact that there is a natural definition of the cohomology
of a space
with coefficients in a sheaf
,
and this without any kind of restrictions on
(this is essential, for example, in algebraic geometry,
where the spaces arising are, as a rule, non-separated)
and to the fact that other cohomologies (under certain
specific conditions) reduce to a sheaf cohomology, at least
in those situations where their application is justified.
To define
one first constructs the
canonical resolution
where
is the sheaf defined by the pre-sheaf
for which
is the group of all (possibly discontinuous) sections of
over
,
so that
,
,
.
By definition,
(
is obtained by replacing the symbol
by
).
The sheaf
itself can be obtained from
so that
(in classical cohomology
is the group of locally constant functions on
with values in
).
The resolution
is an exact covariant functor of
:
there is a short exact sequence of resolutions
corresponding to a short exact
"coefficient"
sequence
.
The functor
turns out to be exact on the terms
,
,
of the resolution, therefore there is an exact cohomology sequence
corresponding to the indicated coefficient sequence, beginning with
.
The cohomology sequence of a pair
corresponds to the short exact sequence,
(
is a closed set).
The cohomology groups
have the following
"universality"
property, which casts light
on their meaning: For any other resolution
(that is, an exact sequence of sheaves
beginning with
)
there is a natural
"comparison"
homomorphism
,
which is described in terms of
by using spectral sequences. An important case is when the sheaves of the resolution are
-acyclic,
that is, when
for
;
in this case the above homomorphism is an
isomorphism. The basic examples of acyclic sheaves are
flabby sheaves
(for all
the mappings
are epimorphic) and
soft sheaves
(any section over a closed set extends to a section over the whole of
).
The canonical resolution consists of flabby sheaves. If
is a paracompact space, then every flabby sheaf is also soft.
The universality property enables one to compare cohomologies arising
in concrete situations with sheaf cohomology (and consequently also with
each other), to discern for them the natural bounds within
which their application is effective, and also to apply sheaf-theoretic
methods to the solution of concrete
problems. For example, Aleksandrov–Čech cohomology (cf.
Aleksandrov–Čech homology and cohomology)
can be defined using cochains obtained from the
cochains of a specially selected system of open coverings by
taking the direct limit. These cochains turn out to be sections
of the sheaves of germs of cochains (defined analogously to the
sheaves of germs of functions) constituting a resolution of the group (or
even the sheaf) of coefficients, and this sheaf turns out to
be soft if the space is paracompact. Thus, for paracompact
spaces Aleksandrov–Čech cohomology coincides with a sheaf cohomology. An
analogous conclusion holds for Zariski spaces (in particular, for algebraic varieties).
Alexander–Spanier cochains
also turn out to be sections of the sheaves of a
resolution and, moreover, the resolution consists of soft sheaves if
is paracompact, so in this case, in particular, Alexander–Spanier and
Aleksandrov–Čech cohomology
are naturally isomorphic. In the case of singular cohomology,
identification of cochains which coincide on the
"small"
singular
simplices, i.e. subordinated to (arbitrary) open coverings, leads to the so-called
localized cochains
(giving the same cohomology), which are sections of the
sheaves determined by the pre-sheaves of the usual singular
cochains. These sheaves turn out to be soft if
is paracompact (if
is hereditarily paracompact, then they are also flabby), but
they form a resolution only under the additional requirement that
is
weakly locally contractible
(in every neighbourhood
of each point
there is a smaller neighbourhood which is contractible to a point inside
).
A classic example is
de Rham's theorem:
The cohomology of the complex of differential
forms of a differentiable manifold coincides with the
usual cohomology with coefficients in the field
of real numbers (the sheaves of germs of differential
forms are soft and form a resolution of
:
Sufficiently near to each point each closed differential form is exact).
There are also resolutions corresponding to any open or locally
finite closed covering and these enable one to compare the cohomology of
with the cohomology of the coverings (the spectral sequences
for coverings). In particular, this gives an isomorphism if
for
for all elements of the covering and
their finite intersections
(Leray's theorem).
Taking the direct limit with respect to open coverings
gives an isomorphism between Aleksandrov–Čech cohomology
and
sheaf cohomology,
even for non-paracompact
,
provided that there are sufficiently many small open sets
in
for which
when
(Cartan's theorem).
This means that the cohomologies
,
used in algebraic geometry, with coefficients in coherent sheaves,
are also isomorphic to the standard sheaf cohomology
.
General constructions ensuring the existence of a comparison
homomorphism enable one to compare also the cohomology
with the hypercohomology
(analogously,
with
)
in case
is any
differential sheaf
(that is, a sheaf in which for any
the composition
is zero) with
acyclic, where the
are the
derived sheaves
of
(these are the quotient sheaves of the kernel by the image in each dimension
).
The corresponding spectral sequences have many applications. Moreover, if
when
,
then
.
For example, if in place of
a sheaf of chains
is taken (the boundary operator lowers the dimension by one, the elements of
are the chains of the pair
,
and the stalk
),
then one obtains the way the homology
depends on all the possible
.
For a manifold,
when
,
and
,
that is,
Poincaré duality
holds. If
is an open or closed subset of a locally compact space
,
then the homology of
is determined by the sections of
with supports in
,
and the homology of the pair
is determined by the sections of the restriction of
to
.
Conversely (and this is also one of the manifestations of Poincaré duality), if
is any flabby resolution for the cohomology, then the restriction of
to
determines the cohomology of
and the sections of
with supports in
determine the cohomology of the pair
.
Since the sheaves
are flabby for manifolds, the homology sequence of the pair
coincides up to an inversion of the numbering with the cohomology sequence of the pair
.
This means that dualities for manifolds, such as
Lefschetz duality
,
are particular cases of Poincaré duality. It
turns out that the duality relations which are not
covered by this scheme are corollaries of Poincaré duality
and the acyclicity of the manifold in certain dimensions.
Just such a situation arises in the case of a continuous mapping
.
A resolution for the cohomology of
determines a certain differential sheaf
on
for which the stalks
are the direct limits of the cohomology groups
with respect to neighbourhoods
of the points
(and for closed mappings
),
where
.
The way
depends on
is described by the Leray spectral sequence of the mapping
(a particular case of this is the spectral sequence of a
Serre fibration).
Acyclic mappings correspond to the case when
vanishes, thus ensuring that the cohomologies of
and
with corresponding coefficients are isomorphic
(Vietoris' theorem
and
its generalizations). The general constructions referred to above also give the
spectral sequence of a mapping and take into account
(along with their cohomological structure) the degree of disconnectedness
of pre-images of points; this is especially effective for
zero-dimensional or finite-to-one mappings (in the case of coverings it
becomes the Cartan spectral sequence). There are
also special spectral sequences in categories of
-spaces
(spaces on which a group
acts).
In sheaf cohomology there is a natural way of defining
a multiplicative structure. The existence of special flabby resolutions, of
which the mappings are determined by a certain semi-simplicial structure,
enables one to give explicit formulas for the products of cochains,
analogous to the usual ones. At the same time this also
makes it possible to define other cohomology operations in sheaf theory.
The apparatus of sheaf theory has many applications
wherever abstract homological methods are essential: in
topology (homological and cohomological dimension, local homology and duality,
the structure of various classes of continuous mappings, including
imbeddings onto dense subsets, and, in particular, compactifications, etc.),
in the theory of analytic manifolds (homology and cohomology
with coefficients in coherent analytic sheaves and their
applications, cohomology and analytic differential forms, homology and analytic
flows (the analogue of de Rham's theorem), etc.), and
also in abstract algebraic geometry (the cohomology of affine,
projective and complete algebraic varieties with
coefficients in coherent algebraic sheaves, algebraic
Serre duality,
algebraic (combinatorial) dimension, etc.).
Some basic notions of sheaf theory and
spectral sequences
appeared in the work of
J. Leray
(1945
and later) in connection
with the study of homological properties of continuous mappings of locally
compact spaces, and he also gave the definition of cohomology (with
compact support) with coefficients in a sheaf. A fairly
complete account of sheaf theory using resolutions was later given by
H. Cartan.
The proof of the de Rham theorem given by
A. Weil
(1947)
and the work of
J.-P. Serre
(in the
early
1950's) on algebraic varieties greatly influenced the development of
sheaf theory. Cohomology with coefficients in a sheaf was first
defined by the Aleksandrov–Čech method. A mature view of sheaf theory could
be found by the end of the
1950's in the work of
A. Grothendieck
[3]
and
R. Godement
[2],
where great generality was achieved and the methods were considerably simplified.
E.g., it was shown that the category of sheaves over
has a
generator
(that is, a sheaf
admitting non-zero homomorphisms into any non-zero sheaf; for sheaves of Abelian groups,
),
and hence that each sheaf can be imbedded
in an injective sheaf
(Grothendieck's theorem).
This is the reason for the formal analogy between cohomology theory
with coefficients in sheaves and the theory of derived functors
in the module categories: In the category of sheaves over
there are
"enough"
injective objects (although, as a rule,
there are few projective objects), and therefore one can
freely apply all the corresponding techniques of homological
algebra; in particular, one can define the cohomology
(without any restriction on
)
as derived functors of the left exact functor
(or even as
).
This also sheds light, for example, on the general
nature of such concepts as the cohomological dimension (over
)
of a space, the algebraic dimension of a variety and the global dimension of
a ring. The description given by Grothendieck of the spectral sequence for the functor
is essential in algebraic geometry. A much
simpler method of constructing injective sheaves was found by
Godement. He also showed that to construct a cohomology theory
it is entirely sufficient to use his canonical flabby
resolution, which, from the point of view of homological algebra,
turns out to be simply one of the acyclic resolutions of
a sheaf. Godement was the first to apply flabby and soft sheaves (soft sheaves are
acyclic only for
paracompact, which explains their use primarily in topology).