Last update: 11 June 2013
We need some basic homological algebra. Let be an abelian category (for our purposes, will be the category of where is a sheaf of rings on a topological space An object in is injective if the functor is exact and not merely left exact: that is to say, whenever is a monomorphism in the map is surjective.
The category has enough injectives if every object in can be embedded in an injective object. Suppose that has enough injectives, and let be an object in Then there exists an injective in and a monomorphrsm Let then there exists an injective in and a monomorphism Let and so on. The short exact sequences etc., then stick together to form a long exact sequence:
called an injective resolution of
Now let be a covariant additive left exact functor on with values in an abelian category If we operate on with we get a complex, so we can form its cohomology:
The central fact is that depends (up to isomorphism) only on and and not on the injective resolution: it is denoted by and is an additive functor, called-the right derived functor of Since is left exact, we have
If is injective then for all for is an injective resolution of
Theorem (8.1). If is an exact sequence in and if is a covariant additive left exact functor on with values in an abelian category then there is an exact sequence in
For the definition of the 'coboundary morphisms' and the proof of (8.1) we refer to Godement's book (or any book on homological algebra).
We shall apply this machinery to the following situation: is a ringed space and is the category of Then is abelian (as remarked in Chapter 7) and in fact has enough injectives (proof e.g. in Godement's book). By (7.2), the section functor is a left exact functor on with values in the category of The cohomology groups (which are in fact of with coefficients in the are then defined to be
In particular, From (8.1) we have an exact cohomology sequence: if is an exact sequence of then the sequence
This definition of the cohomology groups is due to Grothendieck.
There is an earlier definition of sheaf cohomology, modelled on theory, which goes as follows. Let be any open covering of If is any i.e, sequence of elements of the index set let denote the intersection An (alternating) of the covering with coefficients in the sheaf is a function which associates with each an element in such a way that is alternating in the indices and whenever any two of the indices are equal. The form a group which has a natural structure: if then is defined to be If we order the index set linearly then we may write where in the product runs over all such that
Define a coboundary homomorphism
as follows: if then
One verifies that Thus is a complex of and we define the cohomology group of the covering with coefficients in to be
Next one shows that a refinement of gives rise to well-defined homomorphisms with the usual transitivity properties; these enable us to define the cohomology groups of with coefficients in
the direct limit being taken over arbitrarily fine open coverings of
The advantage of cohomology is that one stands some chance of being able to compute it in given situations. The disadvantage, which is a serious one, is that the cohomology sequence in cohomology is not necessarily exact. It is always the case that and (so that the cohomology sequence is always exact as far as but and are not necessarily the same for There is a spectral sequence relating the two cohomologies (details in Godement's book), from which one can assert that for all under suitable hypotheses on or or both. Here are two such 'comparison theorems':
Theorem (8.2). Let be an open covering of let be a sheaf on and suppose that, for all simplexes we have for all Then for all
Theorem (8.3). (Cartan). Let be an open covering of and a sheaf on such that
|(i)||is closed under finite intersections;|
|(ii)||the sets of form a basis of|
|(iii)||for all and all|
Then for all
Theorems (8.2) and (8.3) are proved in Godement's book. We shall sketch a proof of (8.2) avoiding the use of spectral sequences, but not (8.3). There are other comparison theorems: thus the conclusion of (8.3) is valid if is paracompact (and Hausdorff) and is any sheaf of abelian groups. This one is of use if is a differentiable manifold or a complex manifold, but not in algebraic geometry.
Let be a ringed space and an and let be any open covering of For each open set in let denote the open covering of Then we have for each open set in and each integer hence presheaves for each These presheaves are easily verified to be sheaves; denote them by The coboundary operator gives rise to sheaf homomorphisms
also we have a sheaf homomorphism defined as follows: if is a section of over then
Proposition (8.4). The sequence
(i) is a monomorphism. For if then
hence (since the
(ii) Let If then for all pairs in i.e. in hence the fit together to give a section of over such that for each i.e. Conversely, if for some then hence
(iii) We have hence Conversely, let be such that say Then there exists an open neighbourhood of contained in and an element such that If is a let denote the We have hence is a family where runs through the and Define by the rule then
Hence and therefore
|Proof of (8.2).|
(i) Any product of injectives is injective (this is true in any abelian category).
(ii) If is an injective and is open in then is an injective For we have for any where denotes the sheaf on obtained by extending by zero outside
(iii) If is an injective and is the embedding of the open set in then is an injective For we have for any
(iv) With the notation of (8.4), we have where runs through all such that (with respect to some linear ordering of the index set and is the embedding of in Hence if is injective, then is injective by (i), (ii) and (iii).
(v) Let be an injective resolution of Then for each simplex the sequence is an injective resolution of by (iii) and the fact that restriction to an open set preserves exactness. Hence this sequence can be used to calculate the cohomology of But by hypothesis for all Hence the sequence
is exact. Hence, taking the product of these exact sequences for, all the sequence
(vi) Consider next the resolution of
By (8.4) this is an exact sequence. By (iv) above, each is injective, hence this is an injective resolution of but has zero cohomology in dimensions hence the sequence
that is to say the sequence
(vii) We now have a double complex, in which all rows except for the top one, and all columns except for the left-hand one, are exact sequences (by (v) and (vi)):
In such a situation the cohomology of the top row is isomorphic to the cohomology of the left-hand column. But in the present case the cohomology of the top row is the cohomology and the cohomology of the left-hand column is Grothendieck cohomology
This is a typed excerpt of the book "Algebraic Geometry: Introduction to Schemes - I.G. Macdonald".