Sheaf cohomology
Posted by Akhil Mathew under algebra, algebraic geometry | Tags: Cech cohomology,
cohomology, derived functors, sheaf cohomology, sheaves |
[2] Comments
To continue, I am now going to have to use the language of sheaves. For it, and for all details I will omit
here, I refer the reader to Charles Siegel’s post at Rigorous Trivialties and Hartshorne’s Algebraic
Geometry. When I talk about sheaf cohomology, it will always be the derived functor cohomology. I will
briefly review some of these ideas.
Sheaf cohomology
The basic properties of this are as follows.
First, if
is a topological space and
sheaves on
, then
is a covariant additive functor from
to the category of abelian groups. We have
that is to say, the global sections. Also, if
is a short exact sequence of sheaves, there is a long exact sequence
Finally, sheaf cohomology (except at 0) vanishes on injectives in the category of sheaves.
In other words, sheaf cohomology consists of the derived functors of the (left-exact) global section
functor.
When
is flasque (i.e., for any
open with
, restriction
is
surjective), we have
(This is basically because of a well-known fact about sheaves: if
and
is exact
is flasque, then the sequence of global sections is also exact.)
Moreover, this gives a way to compute the cohomology of a sheaf
, then consider the complex
(and
). The key formula is then:
. If we have a flasque resolution
of the global sections of the sheaves
There are many interesting results about the cohomology of a coherent sheaf over schemes, though they
are not really relevant to us now. We are interested in the analytic applications.
Cech cohomology
Cech cohomology gives a reasonable way to actually compute these cohomology groups. For instance,
Hartshorne uses it to calculate the cohomology of line bundles on projective space. In general, Cech
cohomology does not equal derived functor cohomology, though it does in certain cases (in algebraic
geometry, if the scheme is separated, then it is ok).
In the analytic case, we are working with paracompact spaces, so Cech cohomology will always be
acceptable, as we will see.
Let
be a covering of
, and let
be a sheaf on
with boundary maps defined as follows. Let
for every
-tuple
. Consider the chain complex
with corresponding
. Define
where the hat denotes omission, as usual. (Also,
when
.) This can be checked
to be a complex. Then we define the Cech cohomology groups
as the cohomology of this (co)chain complex.
There is another way to interpret the Cech cohomology groups that better illustrates their connection
with regular cohomology. Given
The boundary map
, consider the sheaves
defined by
is defined the same as before, so we have a complex of sheaves
Proposition 1 The above complex is exact.
Exactness at the first step is basically the definition of a sheaf: cocyles in
of elements of
represents collections
that agree on the common intersections, so piece together to a section of
.
Now consider the other more general case. We need to check exactness on the stalks, say at
with
. Let
be such that
on some neighborhood containing
Thus, for all
Then
-tuples
is a section of
to get some
; we can assume
that goes to
via
. Choose
. We need to lift
.
, define
on
. In this way, we get a map from
, and it is easy to check
that it is a chain homotopy, which implies exactness.
Cech cohomology versus derived functor cohomology
I will now discuss some examples of Cech cohomology. The notation remains the same.
Proposition 2
.
This is really just the sheaf axiom; see the beginning of the proof of exactness of the Cech complex of
sheaves.
The next proposition gives us some reason to suspect a relation between tehse two types of cohomology.
Proposition 3
There is a natural transformation of
if
Let
.
-functors
be an injective resolution of
. Then by a basic result in homological algebra, there is a unique
commutative diagram of resolutions
which induces a corresponding commutative diagram on the global sections. Taking the cohomology of
the complex then gives the result.
Next up: the Leray theorem.
Incidentally, I’m not sure how I should categorize this post. I’m using “algebraic geometry” because that
subject seems to depend the most heavily on sheaves.
Sheaf cohomology
Posted by Akhil Mathew under algebra, algebraic geometry | Tags: Cech cohomology,
cohomology, derived functors, sheaf cohomology, sheaves |
[2] Comments
To continue, I am now going to have to use the language of sheaves. For it, and for all details I will omit
here, I refer the reader to Charles Siegel’s post at Rigorous Trivialties and Hartshorne’s Algebraic
Geometry. When I talk about sheaf cohomology, it will always be the derived functor cohomology. I will
briefly review some of these ideas.
Sheaf cohomology
The basic properties of this are as follows.
First, if
is a topological space and
sheaves on
, then
is a covariant additive functor from
to the category of abelian groups. We have
that is to say, the global sections. Also, if
is a short exact sequence of sheaves, there is a long exact sequence
Finally, sheaf cohomology (except at 0) vanishes on injectives in the category of sheaves.
In other words, sheaf cohomology consists of the derived functors of the (left-exact) global section
functor.
When
is flasque (i.e., for any
surjective), we have
open with
, restriction
is
(This is basically because of a well-known fact about sheaves: if
and
is exact
is flasque, then the sequence of global sections is also exact.)
Moreover, this gives a way to compute the cohomology of a sheaf
, then consider the complex
(and
. If we have a flasque resolution
of the global sections of the sheaves
). The key formula is then:
There are many interesting results about the cohomology of a coherent sheaf over schemes, though they
are not really relevant to us now. We are interested in the analytic applications.
Cech cohomology
Cech cohomology gives a reasonable way to actually compute these cohomology groups. For instance,
Hartshorne uses it to calculate the cohomology of line bundles on projective space. In general, Cech
cohomology does not equal derived functor cohomology, though it does in certain cases (in algebraic
geometry, if the scheme is separated, then it is ok).
In the analytic case, we are working with paracompact spaces, so Cech cohomology will always be
acceptable, as we will see.
Let
be a covering of
, and let
with boundary maps defined as follows. Let
for every
-tuple
be a sheaf on
. Consider the chain complex
with corresponding
. Define
where the hat denotes omission, as usual. (Also,
to be a complex. Then we define the Cech cohomology groups
when
.) This can be checked
as the cohomology of this (co)chain complex.
There is another way to interpret the Cech cohomology groups that better illustrates their connection
with regular cohomology. Given
The boundary map
, consider the sheaves
defined by
is defined the same as before, so we have a complex of sheaves
Proposition 1 The above complex is exact.
Exactness at the first step is basically the definition of a sheaf: cocyles in
of elements of
represents collections
that agree on the common intersections, so piece together to a section of
.
Now consider the other more general case. We need to check exactness on the stalks, say at
with
. Let
be such that
on some neighborhood containing
Thus, for all
Then
-tuples
is a section of
to get some
; we can assume
that goes to
via
. Choose
. We need to lift
.
, define
on
. In this way, we get a map from
, and it is easy to check
that it is a chain homotopy, which implies exactness.
Cech cohomology versus derived functor cohomology
I will now discuss some examples of Cech cohomology. The notation remains the same.
Proposition 2
.
This is really just the sheaf axiom; see the beginning of the proof of exactness of the Cech complex of
sheaves.
The next proposition gives us some reason to suspect a relation between tehse two types of cohomology.
Proposition 3
There is a natural transformation of
if
-functors
.
Let
be an injective resolution of
. Then by a basic result in homological algebra, there is a unique
commutative diagram of resolutions
which induces a corresponding commutative diagram on the global sections. Taking the cohomology of
the complex then gives the result.
Next up: the Leray theorem.
Incidentally, I’m not sure how I should categorize this post. I’m using “algebraic geometry” because that
subject seems to depend the most heavily on sheaves.