Fundamentals of (Abelian) Group Cohomology

In this post we will talk about the basic theory of group cohomology, including the cohomology of profinite groups.

We will assume that the reader is familiar with the basic theory of derived functors as in, say, Weibel’s Homological Algebra.

Group Cohomology

The Category of G-Modules

Let be a group. We say that an abelian group is a -module if it comes equipped with a -action which is additive: for all . Of course, as is the case in most parts of mathematics, this is not the only way to define a -module. For example, the above action is clearly equivalent to equipping with a homomorphism , where the automorphisms are group automorphisms.Perhaps less tautological, is the equivalence of -modules with -modules.

Recall that , called the integral group ring over , is the free abelian group on the set , with multiplication given by linearly extending the group operations on . So, for example, if , the group of -roots of unity in , then is the free abelian group on (where is a primitive -roots of unity). The multiplication between the formal symbols is given by

So, in this instance, we see that is given merely by .

Now, the relation between -modules and -modules is simple. Any -module can be made into a -module by merely extending the action of on linearly. Conversely, given a -module , the restriction of the action of the scalars to the subset defines a -module structure.

There is a fancier way to understand this equivalence though. Namely, consider the group ring functor  given on objects by  and given on morphisms by extending linearly:

It turns out that the group ring functor is the left adjoint to the units functor taking . Thus, if is a group, and is a ring, we have that

Thus, if is an abelian group, and is its ring of group endomorphisms, then the above adjunction tells us that

But, a ring map is the same thing as giving a -module structure, and as noted above, a group map is just defining a -module structure on . Thus, we see that these two notions are the same.

The important thing to note about this equivalence is that it tells us that the category of -modules is abelian, being equivalent to the category of (left) -modules. In the future, we shall freely pass between the two equivalent notions of -modules, depending on which is more convenient. We shall denote the morphism sets in either as or depending on which is more illuminating.

Let us now consider some examples of -modules which might be of interest to us:

Every abelian group is a -module with the trivial structure given by for all (i.e. the homomorphism is trivial). We shall call a group with the trivial structure a trivial -module. For all intents and purposes, a trivial -module is just the same as an abelian group.

Let be a Galois extension of fields. Then, (thought of as an additive group) is a -module with the usual action for and . Similarly, becomes a -module with the structure for and . Succinctly, this is because is both a subgroup of and .

In a similar vein, let be a Galois extension of either local or global fields. Then, and become -submodules of and respectively.

Let be an abelian variety over . Then, the group  becomes a module over the absolute Galois group with the usual action on the coordinates.

Let be the integral polynomial ring in -variables. Consider the action of the symmetric group on given by permuting the indeterminates. Clearly acts not only as group automorphism, but in fact ring automorphisms. Thus, we see that with this action becomes an -module.

Let be some object over a field (e.g. a -vector space or a -algebra). Then, , where is a Galois extension, is a -module with .

Every representation one might have seen in a graduate algebra course, in other words a homomorphism , where is some -space, usually , makes into a -module. There are two equivalent ways of thinking about this. The easier of the two is just to note that (where on the right we are just considering with its underlying group structure). Thus, a representation is, in particular, a homomorphism . The other is to recall that giving a representation of on is the same as endowing with a -module structure (this may have even been your definition of -representation. But, this then clearly also endows with a () structure. One can think about representation theory (at least of finite groups) as the study of linearized -modules.

As you can see from the above examples, some of the most important examples of -modules occur in the context of number theory.

 

Group Cohomology: Motivation

Now, while -modules are often useful objects themselves, very often we are more interested in a certain group associated to a -module. In particular, if is a -module, let us denote by the group of invariants of , which is defined as follows:

In other words, is just the set of fixed points of the -action on . Note that while does have an inherited -module structure from , this structure is trivial (it’s the largest such -submodule of ) and so really carries no more structure than the underlying abelian group of .

Usually, the group of invariants of a -module has some sort of geometric or arithmetic meaning. The whole -module is often something large we can get a hand on, and the set of -modules is the object that we’re really after.

Consider the example of an abelian variety over as above. We are often interested in the set of -points, but can usually only get a hand on the full set of -points. But, we can recover as the set of -fixed points of .

To be more concrete, may be an elliptic curve over . The -points of are something we have a somewhat good understanding of. But, what we’re really interested in are the -points, which are nothing but the -fixed points of .

As another example, consider the case discussed above where is a Galois extension of fields, and is a -object. Then, is a -object which might be of a simpler nature. In simple cases, it is easy to show that if acts on as above, then as a -object.

Now, since the full -modules are usually the objects for which we can more easily deal with, we can often times find relationships between them–ways of putting two simpler -modules together to describe a possibly more complicated -module. More rigorously, I mean that we are often times able to find exact sequences of -modules:

But, since it’s really the group of invariants of theses -modules that we may care about, we would really like it to be true that can also be put together nicely from the modules and . Or, said differently, we’d like the induced sequence

to be exact. But, we don’t always get what we want. It turns out that, in general, this sequence need not be exact. In particular, it is always true that the sequence

is exact, but it is not true that taking the invariant module preserves surjections.

As a quick example of this, consider the exact sequence of groups

where denotes the -power map. Note that this is actually an exact sequence of -modules. Passing to the -invariants gives us

where, of course, we have neglected to put a on the right hand side, because the -power map, while surjective on , is not surjective on .

This is where the general ideology of (right) derived functors comes into play. Namely, the functor of -invariants is left exact, but not right exact. We can thus produce it’s associated right derived functors , which we will denote by . The general theory then tells us that for every exact sequence

of -modules, we get the long exact sequence

thus, effectively, fixing the non-surjectivity by adding in the groups , and . These right derived functors, will be what we define as the group cohomology functors.

Group Cohomology: Definitions and Basic Theorems

So, now that we have (hopefully) been sufficiently motivated for the definition of group cohomology, we can dive right into it. As stated above, the fixed points functor is a left exact (necessarily additive!) functor from the abelian category to . Since is equivalent/isomorphic to the category , and categories of modules always have enough injectives, we know that also has enough injectives.

Thus, we are able to form the right derived functors . If is a -module, we then define the  (group) cohomology of with -coefficients, denoted , to be .

Just to recall, here is the general methodology to produce . First we find a resolution of by injective -modules (which we know exists by the above discussion), we then obtain as the homology group of the deleted complex .

Now, while this presentation of is the most obvious in definition, there is an equivalent, but notationally different, way of defining . This different presentation comes from the realization that the functor is naturally isomorphic to another functor for which we already have well-understood theory of its derived functor. Namely, let have the trivial -module structure, and define then:

Theorem 1: The two functors and are isomorphic.

Proof: Define the morphism as follows. For any -module , define

To see that this is indeed well-defined (i.e. that ), we merely note that for any  we have that

which does, in fact, show that . Now, it’s clear that is a morphism of abelian groups. We claim that it is an isomorphism. The injectivity follows immediately from the fact that generates as a -module. For surjectivity, let be arbitrary. Define by . A quick check shows that is a -module map, and that , giving the desired surjectivity of .

It thus remains to check the coherence property of . Given -modules and , and a -module map we need to check that

But, if , then the top and the right arrow give me and the left and then bottom arrow give me and so commutativity is proven.

Now, while on the face this just seems neat “our functor is just a Hom functor!” it actually has a much more consequential corollary:

Corollary 2: Let be a group. Then, there are natural isomorphisms .

Proof: This just follows since the functors and are isomorphic, and so for every

as desired.

This is great since the theory of the functor is very well understood, all of which we can now bring to bear on our study of group cohomology!

 

Group Cohomology: The Bar Resolution

One of the biggest advantages that Corollary 2 gives us, is the ability to create a very concrete description of the cohomology groups –in particular . The reason for this is simple. Now that we know that we have been granted a huge computational boon.

As of now, our technique for computing has been to find an injective resolution for , take the deleted resolution, apply , and then take homology. With this approach, finding a uniform description of sounds very difficult since there is no uniform injective resolution that works for all (duh!).

But, now that we know we see that we now have a fighting chance of getting our desired uniform description. Indeed, while it seems fruitless to try and describe some universally nice injective resolution we know that to find we can either find an injective resolution of , or a projective resolution of . In particular, we see that instead of having to find some general resolution which is amenable for all possible , we can find one resolution for , and see how the mash up against it.

This is where the bar resolution comes in. It’s a fairly canonical projective resolution which will be conducive to giving a straightforward presentation of the ‘s.

Let us define to be the free -module on , where we give the structure of a -module via extending linearly the “diagonal action” . We claim that is actually a free -module. More specifically:

Theorem 3: Let be as above. Then,  is a basis for as a -module.

Proof: To see that they are -independent, suppose that

then we’d have that

But, note that is an injection, and thus we see that this is a linear coefficients of -basis elements which gives . Thus, each must be zero, and so each coefficient

as desired.

To see that is a spanning set of , it suffices to show each basis element is a -linear combination of elements of this set. But, note that

from where the conclusion follows.

With this theorem proven, we can unabashedly define to be the fundamental basis for .

As the notation suggests, the will be the terms of our bar resolution . So, we need to construct differentials . We do so by extending -linearly the following

where, as usual, the hat denotes leaving off that element of . One can easily check that this is, in fact, a map of -modules.

Now, verifying that is very similar to the general such computation associated to simplicial complexes (e.g. the one for simplicial homology in algebraic topology)–one just makes a rearrangment of the terms so full cancellation occurs.

So, now we can finally define the bar resolution of with respect to to be the following resolution:

where is the so-called augmentation map given by

We have the following claim which, if there is any sense in the phrase “bar resolution”, is true:

Theorem 4: The bar resolution is a resolution.

The proof is elementary, and is available in any text containing a chapter on group cohomology. The main idea being to construct a null-homotopy of the resolution.

So, now that we know that really is a projective resolution of as a -module, we know that

So, this is not very useful, unless we can nicely describe the complex

But, of course, since we wouldn’t be talking about the bar resolution unless we could have slick description, we can infer that we must. There are two common explications of this above sequence. We give the first mostly for cultural reasons since it is occasionally mentioned, but it is the second that will be the real workhorse for us.

 

For the first construction, let us define the homogenous cochains associated to the pair as the group

In other words, an homogenous cochain is just a (set!) map which commutes with the diagonal action of on . Of course, is a group with

which, as one can check, really does define another element of .

To see why we care about these homogenous cochains, let’s consider what an element of looks like. Well, since is generated as a -module by we know that we can identify with its values on . In this way we obtain an injection

which is easily seen to be a homomorphism. The question remains as to what the image of this map is. Since is a -module map, we know that for any and we have that

Thus, we see that the image of lies in . Conversely, if then we see that

and so is the -linear extension of a map , which we always know does give a well-defined -map since is a basis.

In this way, we obtain group isomorphisms

from which we define the maps to be such that the following ladder diagram commutes:

A moment’s thought shows that , for has the following explicit description

as one would expect.

 

Now, let us describe the second, more practical identification of . The basic idea is that while above we identified an element of with its restriction to the generating set , it is much more natural (as was even witnessed above by some comments) to identify with its restriction to the basis .

To this end, let us define the  inhomogenous cochains of the pair to be the group

where, as above, has the group structure by adding maps pointwise. Since is a basis for as a -module we obtain isomorphisms

where the last isomorphism comes from the tautological bijection .

We define differentials , as before, to be such that

commutes. Now, while it is less obvious than the case what does explicitly, a quick computation shows that

This shows the advantages of the homogenous cochains over the inhomogenous. While the groups of inhomogenous cochains are much simpler than their homogenous counterpart, this is made in exchange for a considerably more complicated differential. It turns out though that, for explicit computations, it is the simpler groups (i.e. the inhomogenous cochains) that we shall want.

 

An Explicit Description of the First Cohomology Group

Now, from the construction of homogenous and inhomogenous cochains, and their differentials, we have the following:

Theorem 5: Let be a group, and a -module. Then, for all

The most important consequence of  Theorem 5 is a very concrete way to deal with .

To make this description, all we need to do is think about what an element of and an element of look like. Well, given an element , which is just a map , we know that

So, if and only if for all we have that

where, it is important to note, that the addition is taking place in (i.e. it’s ‘s addition), but the multiplication is the -action on .

We call an element of a crossed homomorphism . This is because if happens to be a trivial -module, then a crossed homomorphism is just a group homomorphism, and otherwise it’s like a homomorphism that has been “twisted” (or crossed) by the -action. It is worth noting that any crossed homomorphism must take the identity element to the identity element since

which, by subtraction, gives the desired result.

Now, we have left to describe . Well, an element is nothing but a map or equivalently, since , an element . So, we can identify with itself. Then, applying to this element we get

Thus, the elements of are just the maps in defined by for some fixed . We call an element of a principal crossed homomorphism. Principal since it is obtained from just one element .

Thus, Theorem 5 allows us to conclude the following:

Theorem 6: Let be a group, and a -module. Then,

Now, as was mentioned above, if is a trivial -module, then a crossed homomorphism is just a homomorphism. And, a principal crossed homomorphism is just the trivial map since in this case

Thus, we have the following corrollary:

Corollary 7: If is a group, and is a trivial -module, then

 

This explicit characterization also allows us to nicely calculate for any -module .

To this end, let be a finite group, a -module, and let us define the norm map by

Note that actually sends into by Cayley’s theorem. In some sense, is the “freest” possible way to turn a generic element of into a -invariant one. This “averaging” technique should be very familiar to anyone who has studied representation theory.

With this definition of the norm map, we can now give a different, more useful description of . Because there are possible confusions between elements of and generic elements of a group (e.g. vs. !) we shall write . This notation is suggestive of future theorems as well (see Hilbert’s Theorem 90):

Theorem 8: Let be a -module. Then,

Proof: From Theorem 6, we know that is crossed homomorphisms modulo principal crossed homomorphisms. The basic idea is that any principal crossed homomorphism should entirely be determined by its image on . Indeed, while may not be a homomorphism, the crossed homomorphism property implies that

and

and more generally

Thus, we see that the map

is injective. We also see that

and so the image of lands inside of . Conversely, if , then

does define a crossed homomorphism which maps, under , to . Thus, we see that is an isomorphism. Thus, we may conclude that

But, under , a principal homomorphisms maps to . Thus,

and so the theorem follows.

 

Coinduced Modules and Shapiro’s Lemma

We now move on to one of the most important computational, as well as theoretical, tools for group cohomology. The notion of the coinduced module. It will be a pivotal piece of machinery for moving between group cohomology over different groups.

The basic idea of coinduced modules is simple. Suppose that is a group, and a subgroup. We want a way to associate to an -module , some -module which is “freest” in some sense. While there are many possible definitions of “freest”, perhaps the most natural given the context would be that the cohomology does not change. More precisely, that in some natural way. So, the obvious question now is how to construct this from our given .

One way one might achieve constructing such a , is asking something even stronger than just . Note first that is a free -module with basis any transversal of . Thus, any projective resolution of as a -module, is also a projective resolution of as a module. We can then further impose that be such that any time is a projective resolution of as a -module that

where this is an isomorphism of complexes. This then would imply that since

as desired.

Now, if we want this stronger condition to hold, it seems feasible that we’d want

for any -module . But, by Yoneda’s lemma, this uniquely characterizes . So, any module we can find which satisfies this property will be our desired module. One way of getting such a module is to recall the Tensor-Hom Adjunction, which says that if is a ring map, then for any -module and -module we have

So, considering the ring map , the -module , and the -module , we have

and so our desired module is !

So, let us, now not so mysteriously, define for a -module , the coinduced module, denoted , to be the -module . We give the -module structure, as usual, by defining for all .

 

Remark: It’s important to point out that there is another, dual, notion to the coinduced module called the induced module. It’s uses come when one is trying to study group homology. It’s definition, also dual to ours, is just . It turns out that if is of finite index in , then as -modules. For this reason, and perhaps because “coinduced” is less pleasant to say than “induced”, some people elect to call what we call the coinduced module, the induced module. This can be confusing if one is not cognizant of the difference.

 

So, now that we have defined the coinduced module, let’s show that it does, indeed, satisfy the desired property (although, a proof is contained in the above motivaion):

Theorem(Shapiro’s Lemma): Let be a group, a subgroup, and a -module. Then, for all

Proof: Let’s first verify the case . Which, amounts to showing that

But, this is simple if we use our alternate description of the fixed submodule and the universal property which motivated us to define the coinduced module

as desired.

Now, for the general case. Let be an injective resolution of as a -module. We claim then that is an injective resolution of . The fact that this sequence is exact follows from the fact that , which is an exact functor since is a free -module. To see that the terms of are injective, we merely note that for each

which, by the injectivity of , implies that is an exact functor, which implies that is injective.

We thus finish by noting that

which is what we wanted to prove.
One important consequence of Shapiro’s lemma is the acyclicity of so called coinduced modules. For a group , we call a -module  coinduced, without reference to any subgroup, if it is of the form for some -module (i.e. abelian group) . For notational convenience, we shall denote by just .

There is a more natural way to describe the modules though. Namely, by definition is just . Now, since is a free -module, this shows that, as a group, is nothing more than (i.e. a direct sum of ‘s, indexed by the elements of ). The -action on is nothing more than permuting the coordinates so that if is an element of , then is the tuple of whose coordinate in the position is just . Moreover, it’s clear that every -module of this form, for some abelian group is coinduced, being .

We now make precise what we mean by saying that coinduced modules are acyclic:

Theorem 9: Let be a group, and let be an abelian group. Then,

for all .

Proof: By Shapiro’s lemma we know that

But, is trivial. To see this, note that , and so

since is a free -module, and .

 

One nice thing that Theorem 9 allows us to discuss is the notion of weakly injective modules. In particular, if is a group, we say that a -module is weakly injective if it is direct summand (as a -module!) of a coinduced module. But, since cohomology is additive, we may conclude by Theorem 9 that any weakly injective module is acyclic. Thus, we may use weakly injective modules as resolutions to compute cohomology.

Now, note that any coinduced module is also a coinduced module for any subgroup since

where is any set of coset representatives of . From this, we may conclude that every weakly injective -module is a weakly injective -module. But, every injective -module is weakly injective.

To see this, first let us note that every -module can be embedded into a coinduced module. In fact, one can easily check that if is any -module then the mapping

is an embedding of -modules, where we consider just as an abelian group in . Thus, if we consider the short exact sequence of -modules, where is injective,

injectivity says that this sequence splits and so

showing that injective -modules are, in fact, weakly injective.

So, from our above discussion, we see that every injective -module is weakly injective, and so acyclic as an -module. Thus, we know that if is a resolution of a -module by injective -modules, then it is also a resolution of , thought of as an -module, by acyclic -modules. Thus, we may conclude the following:

Theorem 10: Let be a group and a subgroup. If is a -module, and is a resolution of by injective -modules, then

While this doesn’t seem very important, it’s actually very computationally nice. If you want to find the cohomology of some -module , with coefficients in some various subgroups, you only have to find one injective resolution of , which then will serve as a resolution with which you can compute cohomology with coefficients in any subgroup.

 

Functorial Properties of Cohomology: Change of Groups

Now, built into the theory of derived functors is, well, the functorality. In particular, if is a group, and and are two -modules, then any -module map induces group maps for all . One may ask though if is also functorial in the other entry. In particular, one might wonder if given a group map , whether this induces, in some way, group maps on cohomology groups.

To make any sense of this statement, we won’t just need a group map , but a map , where is a -module, and is a ‘-module which, in some way, is coherent with . Rigorously, this means that

for any and . If satisfy this identity, we say that they are an intertwining pair.

Now, the point of an intertwining pair is that it allows us to induce a chain map  inhomogenous cochains of to the inhomogenous cochains of . Let us denote the induced map given by by . Then, our desired chain map is as follows:

where, of course, the on the top and bottom rows are in accordance to their respective pairs or , and the are defined as follows:

noting, of course, that really is a map . The verification that this is a chain map is elementary, and uses, in a pivotal way, the fact that is an intertwining pair. Now, since is a chain map, we know that we get an induced map on cohomology as we wanted.

The name of the game now is just to define, in several situations, natural intertwining pairs and discuss some if their properties:

 

First, let us consider a group , along with a subgroup , and a -module . We may then, of course, also consider as an -module. Let’s then consider the intertwining pair where is inclusion, and is the identity map. By the above, we obtain morphisms

called the restriction map. One can check that with respect to cohomology computed via the bar resolution, the restriction maps take a particularly simple, and name-worthy form. Namely, for a cocycle in , one can check that maps to the cocycle in .

 

Next, we claim that we can describe the isomorphism given to us by Shaprio’s Lemma by means of intertwining pairs. Let us first note that there is a natural map given by taking to . We claim that this map, along with being the inclusion, gives an intertwining pair. But, this ultimately just follows from the fact that each is -linear, because then for any and we have that

 Thus, really is an intertwining pair, and so we get an induced map

We claim that this is the map coming from Shapiro’s Lemma. This is elementary though, and is left for you to check.

 

With this alternate way of describing Shapiro’s Lemma we can alternately describe the restriction maps now. Namely, consider the composition

where in the second and third spots is thought about as an -module, the first map is given by , and the second map is the map described in the previous example. The first map is clearly a map of -modules, and so induces a map

and the second map, just being the one coming from Shapiro’s Lemma, induces an isomorphism

and so the composition of these maps gives rise to a morphism

But, note that the composition is defined by

and so the composition is also just the identity map on , and so we recover that the composition

is just the restriction map .

 

We next describe the so-called “inflation maps”. Let’s suppose that is a group, a normal subgroup, and a -module. Note that becomes a -module by

for any and . It is precisely because all elements of are fixed by that this action is well-defined. Consider now the pair where is the usual projection map, and is the obvious inclusion. We claim that is an intertwining pair. But, this amounts to the assertion that

for any and . Thus, we obtain morphisms

for all , called the inflation maps.

 

The last important case of a “change of groups” morphism, comes as a sort of dual to the restriction maps. Suppose that is a group, and a subgroup of finite index. Let be a -module. Let’s define a map

This map is defined by taking a map to the averaged map

where are coset representatives–one can easily check by reindexing, and the fact that is -linear, that is independent of the choice of coset reps. One also checks that is also -linear as claimed. The fact that this is a morphism of -modules is apparent, and thus we obtain morphisms

these maps are called the corestriction maps.

 

One of the lowest hanging, but still juicy, fruit that comes from these definitions is the following seemingly innocuous theorem:

 

Theorem 11: Let be a group, and a subgroup of index . Then,  is multiplication by , for all .

Proof: The map is described as being the map induced by the composition

as described above, followed by Shapiro’s lemma. The map is defined as being one induced by a map , followed by Sharpiro’s lemma (but the one in the other direction–the inverse of the one in the previous sentence). Thus, the composition is just going to be the one induced by the composition

But, thinking of As this just takes to which, since is -linear, is not just . Thus, the composition is just multiplication by , and since cohomology is additive, the induced map is also just multiplication by .

 

The utility of this theorem is the following corollary:

Corollary 12: If is a finite group of order , then is -torsion for all -modules , and all . In particular, if is a finitely generated group, then is finite.

Proof: For we know that by Shapiro’s lemma. Thus, must be the zero map, having zero target. But, then

as desired.

For the finiteness of , merely note that by construction of the (via the bar resolution, say) the fact that is a finitely generated group implies the same for . But, then is a finitely generated torsion group, and so must be finite.

 

This corollary allows us to also make some immediate computations of cohomology groups. For example, if is -divisible group (i.e. multiplication by is an automorphism ), then for any group of order . Indeed, since is an isomorphism, it induces the isomorphism . But, since is -torsion, this implies that must be zero.