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Notes on class field theory

Kiran S. Kedlaya

Section 3.2 Cohomology of finite groups II: concrete nonsense

Reference.

[36], II.1.
In the previous chapter, we associated to a finite group \(G\) and a (right) \(G\)-module \(M\) a sequence of abelian groups \(H^i(G, M)\text{,}\) called the cohomology groups of \(M\text{.}\) (They’re also called the Galois cohomology groups because in number theory, \(G\) will invariably be the Galois group of some extension of number fields, and \(A\) will be some object manufactured from this extension.) What we didn’t do is make the construction at all usable in practice! This time we will remedy this.

Subsection Induced \(G\)-modules

In light of Exercise 5, to compute cohomology we are going to need an ample supply of acyclic \(G\)-modules. We will get these using a process known as induction.

Remark 3.2.1.

By way of motivation, we note first that if \(G\) is the trivial group, every \(G\)-module is acyclic: if \(0 \to M \to I^0 \to I^1 \to \cdots\) is an injective resolution, taking \(G\)-invariants has no effect, so \(0 \to I^0 \to I^1 \to \cdots\) is still exact except at \(I^0\) (where we omitted \(M\)).

Definition 3.2.2.

If \(H\) is a subgroup of \(G\) and \(M\) is an \(H\)-module, we define the induction of \(M\) from \(H\) to \(G\) to be \(\Ind^G_H M = M \otimes_{\ZZ[H]} \ZZ[G]\text{.}\) We may also identify \(\Ind^G_H M\) with the set of functions \(\phi\colon G \to M\) such that \(\phi(gh) = \phi(g)^h\) for \(h \in H\text{,}\) with the \(G\)-action on the latter being given by \(\phi^g(g') = \phi(gg')\text{:}\) namely, the element \(m \otimes [g] \in M \otimes_{\ZZ[H]} \ZZ[G]\) corresponds to the function \(\phi_{m,g}\) taking \(g'\) to \(m^{gg'}\) if \(gg' \in H\) and to 0 otherwise.

Proof.

The key points are:
  1. \((\Ind^G_H N)^G = N^H\text{,}\) so there is an isomorphism for \(i=0\) (this is most visible from the description using functions);
  2. the functor \(\Ind^G_H\) from \(H\)-modules to \(G\)-modules is exact (that is, \(\ZZ[G]\) is flat over \(\ZZ[H]\text{;}\) in fact it is free over \(\ZZ[H]\));
  3. if \(I\) is an injective \(H\)-module, then \(\Ind^G_H(I)\) is an injective \(G\)-module. This follows from the existence of a canonical isomorphism \(\Hom_G(M, \Ind^G_H I) = \Hom_H(M, I)\text{,}\) for which see Proposition 3.2.6 below.
Now take an injective resolution of \(N\text{,}\) apply \(\Ind^G_H\) to it, and the result is an injective resolution of \(\Ind^G_H N\text{.}\)

Definition 3.2.4.

We say \(M\) is an induced \(G\)-module if it has the form \(\Ind^G_{1} N\) for some abelian group \(N\text{,}\) i.e., it can be written as \(N \otimes_\ZZ \ZZ[G]\text{.}\) (The subscript 1 stands for the trivial group, since \(G\)-modules for \(G = 1\) are just abelian groups.)

Proof.

To complete the previous argument, we need an important property of induced modules. This is closely related to the Frobenius reciprocity law in the theory of representations of finite groups.

Proof.

To begin with, note that if we take \(N = M\) (or more precisely, \(N\) is a copy of \(M\) with only the action of \(H\) retained), then the identity map between \(M\) and \(N\) is supposed to correspond both to a homomorphism \(\Ind^G_H M \to M\) and to a homomorphism \(M \to \Ind^G_H M\text{.}\) Let us write these maps down first: the map \(\Ind^G_H M \to M\) is
\begin{equation*} \sum_{g \in G} m_g \otimes [g] \mapsto \sum_{g \in G} (m_g)^g\text{,} \end{equation*}
while the map \(M \to \Ind^G_H M\) is
\begin{equation*} m \mapsto \sum_i m^{g_i} \otimes [g_i^{-1}] \end{equation*}
where \(g_i\) runs over a set of left coset representatives of \(H\) in \(G\text{.}\) This second map doesn’t depend on the choice of the representatives; consequently, for \(g \in G\text{,}\) we can use the coset representatives \(gg_i\) instead to see that
\begin{equation*} m^{g} \mapsto \sum_{i} m^{gg_i} \otimes [g_i^{-1}] = \left( \sum_{i} m^{gg_i} \otimes [(g g_i)^{-1}] \right)[g]. \end{equation*}
This means that we do in fact get a map compatible with the \(G\)-actions. (Note that the composition of these two maps is not the identity! For more on this point, see the discussion of extended functoriality in Section 3.3.)
Now let \(N\) be general. Given a homomorphism \(M \to N\) of \(H\)-modules, we get a corresponding homomorphism \(\Ind^G_H M \to \Ind^G_H N\) of \(G\)-modules, which we can then compose with the above map \(M \to \Ind^G_H M\) to get a homomorphism \(M \to \Ind^G_H N\) of \(G\)-modules. We thus get a map
\begin{equation*} \Hom_H(M,N) \to \Hom_G(M, \Ind^G_H N); \end{equation*}
to get the map in the other direction, start with a homomorphism \(M \to \Ind^G_H N\text{,}\) identify the target with functions \(\phi\colon G \to N\text{,}\) then compose with the map \(\Ind^G_H N \to N\) taking \(\phi\) to \(\phi(e)\text{.}\)
In the other direction, given a homomorphism \(N \to M\) of \(H\)-modules, we get a corresponding homomorphism \(\Ind^G_H N \to \Ind^G_H M\) of \(G\)-modules, which we can then compose with the above map \(\Ind^G_H M \to M\) to get a homomorphism \(\Ind^G_H N \to M\) of \(G\)-modules. We thus get a map
\begin{equation*} \Hom_H(N,M) \to \Hom_G(\Ind^G_H N,M); \end{equation*}
to get the map in the other direction, start with a homomorphism \(\Ind^G_H N \to M\) of \(G\)-modules and evaluate it on \(n \otimes [e]\) to get a homomorphism \(N \to M\) of \(H\)-modules.

Remark 3.2.7.

Proposition 3.2.6 asserts that the restriction functor from \(G\)-modules to \(H\)-modules and the induction functor from \(H\)-modules to \(G\)-modules form a pair of adjoint functors in both directions. This is rather unusual; it is far more common to have such a relationship in only one direction. Indeed, without assuming that \(G\) is finite (or at least that \([G:H] < \infty\)), then the proof of Proposition 3.2.6 only shows that \(\Hom_G(\Ind^G_H N, M) \cong \Hom_H(N,M).\)

Remark 3.2.8.

The point of all of this is that it is much easier to embed \(M\) into an acyclic \(G\)-module than into an injective \(G\)-module: use the map \(M \to \Ind^G_1 M\) constructed in Proposition 3.2.6! An immediate consequence is that if \(M\) is finite, it can be embedded into a finite acyclic \(G\)-module, and thus \(H^i(G,M)\) is finite for all \(i\text{.}\)
However, contrary to what you might expect, for fixed \(M\text{,}\) even if \(M\) is finite, the groups \(H^i(G,M)\) do not necessarily become zero for \(i\) large. We’ll see explicit examples in the next section.
Another consequence is the following result. (The case \(i=1\) was stated previously in Exercise 2.)

Proof.

Put \(G = \Gal(L/K)\text{.}\) The normal basis theorem (see Lang, Algebra or Milne, Lemma II.1.24) states that there exists \(\alpha \in L\) whose conjugates form a basis of \(L\) as a \(K\)-vector space. This implies that \(L \cong \Ind^{G}_{1} K\text{,}\) so \(L\) is an induced \(G\)-module and so is acyclic.

Subsection Group cohomology via homogeneous cochains

Now let’s see an explicit way to compute group cohomology.

Definition 3.2.10.

Given a group \(G\) and a \(G\)-module \(M\text{,}\) define the \(G\)-module \(N^i\) for \(i \geq 0\) as the set of functions \(\phi\colon G^{i+1} \to M\text{,}\) with the \(G\)-action
\begin{equation*} (\phi^g)(g_0, \dots, g_i) = \phi(g_0g^{-1}, \dots, g_ig^{-1})^g\text{.} \end{equation*}
Notice that this module is induced: we have \(N^i = \Ind_{1}^G N^i_{0}\) where \(N^i_{0}\) is the subset of \(N^i\) consisting of functions for which \(\phi(g_0, \dots, g_i) = 0\) when \(g_0 \neq e\text{.}\)

Definition 3.2.11.

Define the map \(d^i\colon N^i \to N^{i+1}\) by
\begin{equation*} (d^i \phi)(g_0, \dots, g_{i+1}) = \sum_{j=0}^{i+1} (-1)^j \phi(g_0, \dots, \widehat{g_j}, \dots, g_{i+1})\text{,} \end{equation*}
where the hat over \(g_j\) means you omit it from the list.

Proof.

Left to the reader.

Definition 3.2.13.

By Corollary 3.2.5 and Lemma 3.2.12, the sequence
\begin{equation*} 0 \to M \to N^0 \to N^1 \to \dots \end{equation*}
is an acyclic resolution of the \(G\)-module \(M\text{.}\) Hence the cohomology of the complex
\begin{equation*} 0 \to N^{0G} \to N^{1G} \to \cdots \end{equation*}
coincides with the cohomology groups \(H^i(G, M)\text{.}\) And now we have something we can actually compute!
Some terminology: the elements of \(N^{1G}\) are called (homogeneous) \(i\)-cochains. The cocycles in the kernel of \(d^i\) are called (homogeneous) \(i\)-cocycles. The ones in the image of \(d^{i-1}\) are called \(i\)-coboundaries. (This terminology makes little sense here; it is transferred from the classical theory of homology of topological spaces, where it has some geometric significance.)

Subsection Fun with \(H^1\)

Remark 3.2.14.

Using the resolution by homogeneous cochains, we can give a very simple description of \(H^1(G,M)\text{.}\) Namely, a 1-cocycle \(\phi\colon G^2 \to M\) is determined by \(\rho(g) = \phi(e, g)\text{,}\) which by \(G\)-invariance satisfies the relation
\begin{align*} 0 \amp = (d^1\phi)(e, h, gh)\\ \amp = \phi(h, gh) - \phi(e, gh) + \phi(e, h)\\ \amp = (\phi^h)(h,gh) - \rho(gh) + \rho(h)\\ \amp = \phi(e, g)^h - \rho(gh) + \rho(h)\\ \amp = \rho(g)^h + \rho(h) - \rho(gh). \end{align*}
It is the image of a 0-cochain \(\psi\colon G \to M\) if and only if
\begin{equation*} \rho(g) = \phi(e,g) = \psi(g) - \psi(e) = \psi(e)^g - \psi(e). \end{equation*}
That is, \(H^1(G,M)\) consists of crossed homomorphisms modulo principal crossed homomorphisms, consistent with the definition we gave in Section 1.2.

Definition 3.2.15.

We may also interpret \(H^1(G,M)\) as the set of isomorphism classes of principal homogeneous spaces of \(M\text{.}\) Such objects are sets \(A\) with both a \(G\)-action and an \(M\)-action, subject to the following restrictions:
  1. for any \(a \in A\text{,}\) the map \(M \to A\) given by \(m \mapsto m(a)\) is a bijection;
  2. for \(a \in A\text{,}\) \(g \in G\) and \(m \in M\text{,}\) \(m(a)^g = m^g(a^g)\) (i.e., the \(G\)-action and \(M\)-action commute).
To define the associated class in \(H^1(G,M)\text{,}\) pick any \(a \in A\text{,}\) take the map \(\rho\colon G \to M\) characterized by \(\rho(g)(a) = a^g\text{,}\) and let \(\phi\) be the 1-cocycle with \(\phi(e, g) = \rho(g)\text{.}\) The verification that this defines a bijection is left to the reader.

Example 3.2.16.

The identity in \(H^1(G,M)\) corresponds to the trivial principal homogeneous space \(A=M\text{,}\) on which \(G\) acts as it does on \(M\) while \(M\) acts by translation: \(m(a) = m+a\text{.}\)

Remark 3.2.17.

This interpretation of \(H^1\) appears prominently in the theory of elliptic curves.
  1. For example, if \(L\) is a finite extension of \(K\) and \(E\) is an elliptic curve over \(K\text{,}\) then \(H^1(\Gal(L/K), E(\bar{K}))\) is the set of \(K\)-isomorphism classes of curves whose Jacobians are \(K\)-isomorphic to \(E\) and which have an \(L\)-rational point but not necessarily a \(K\)-rational point. For any such curve \(C\text{,}\) we can define the translation map \(E \times_K C \to C\) by first defining it over \(L\text{,}\) by picking some \(L\)-rational point to use as the origin, then observing that the result is independent of the chosen point.
  2. For another example, \(H^1(\Gal(L/K), \Aut(E_{\bar{K}}))\) parametrizes twists of \(E\text{,}\) elliptic curves defined over \(K\) which are \(L\)-isomorphic to \(E\text{.}\) (E.g., \(y^2 = x^3+x+1\) versus \(2y^2 = x^3 + x + 1\text{,}\) with \(L = \QQ(\sqrt{2})\text{.}\)) In this example the translation action is not so obvious, and its existence depends on the fact that \(\Aut(E_{\bar{K}})\) is abelian. (One can interpret twists similarly for more general curves, for which the automorphism group need not be abelian, but then \(H^1\) won’t make sense the way we have defined it; it will only have the structure of a pointed set.)
See [51], especially Chapter X, for all this and more fun with \(H^1\text{,}\) including the infamous Selmer group and Tate–Shafarevich group.

Subsection Fun with \(H^2\)

Remark 3.2.18.

We can also give an explicit interpretation of \(H^2(G,M)\) (see [36], example II.1.18(b)). It classifies short exact sequences
\begin{equation*} 1 \to M \to E \to G \to 1 \end{equation*}
of (not necessarily abelian) groups on which \(G\) has a fixed action on \(M\text{.}\) The action is given as follows: given \(g \in G\) and \(m \in M\text{,}\) choose \(h \in E\) lifting \(G\text{;}\) then \(h^{-1}mh\) maps to the identity in \(G\text{,}\) so comes from \(M\text{,}\) and we call it \(m^g\) since it depends only on \(g\text{.}\)
The correspondence is constructed as follows. Given an exact sequence as above, choose a map \(s\colon G \to E\) (which need not be a homomorphism) such that \(s(g)\) maps to \(g\) under the map \(E \to G\text{.}\) Then the map \(\phi\colon G^3 \to M\) given by
\begin{equation*} \phi(a,b,c) = s(a)^{-1} s(ba^{-1})^{-1} s(cb^{-1})^{-1} s(ca^{-1}) s(a) \end{equation*}
is a homogeneous 2-cocycle, and any two choices of \(s\) give maps that differ by a 2-coboundary.
What “classifies” means here is that two sequences give the same element of \(H^2(G,M)\) if and only if one can find an arrow \(E \to E'\) making the diagram in Figure 3.2.19 commute.
Figure 3.2.19.
Note that two sequences may not be isomorphic under this definition even if \(E\) and \(E'\) are abstractly isomorphic as groups. For example, if \(G = M = \ZZ/p\ZZ\) and the action is trivial, then \(H^2(G, M) = \ZZ/p\ZZ\) even though there are only two possible groups \(E\text{,}\) namely \(\ZZ/p^2\ZZ\) and \(\ZZ/p\ZZ \times \ZZ/p\ZZ\text{.}\)

Remark 3.2.20.

One can similarly interpret \(H^i(G,M)\) for \(i > 2\) in terms of longer exact sequences; this is similar to the construction of higher Yoneda extension groups. See [22].

Subsection Extended functoriality

We already saw that if we have a homomorphism of \(G\)-modules, we get induced homomorphisms on cohomology groups. But what if we want to relate \(G\)-modules for different groups \(G\text{,}\) as will happen in our study of class field theory? It turns out that in a suitable sense, the cohomology groups are also functorial with respect to changing \(G\text{.}\)

Definition 3.2.21.

Let \(M\) be a \(G\)-module and \(M'\) a \(G'\)-module. Suppose we are given a homomorphism \(\alpha\colon G' \to G\) of groups and a homomorphism \(\beta\colon M \to M'\) of abelian groups (note that they go in opposite directions!). We say these are compatible if \(\beta(m^{\alpha(g)}) = \beta(m)^g\) for all \(g \in G\) and \(m \in M\text{.}\) In this case, one gets canonical homomorphisms \(H^i(G, M) \to H^i(G', M')\text{:}\) one firsts constructs these on pairs of injective resolutions, then shows that any two choices are homotopic and hence give the same maps on cohomology. We will refer to the construction of such homomorphisms as the extended functoriality of group cohomology.

Example 3.2.22.

The principal examples of extended functoriality we will be using are the following.
  1. Note that cohomology groups don’t seem to carry a nontrivial \(G\)-action, because you compute them by taking invariants. This can be reinterpreted in terms of extended functoriality: let \(\alpha\colon G \to G\) be the conjugation by some fixed \(h\text{:}\) \(g \mapsto h^{-1}gh\text{,}\) and let \(\beta\colon M \to M\) be the map \(m \mapsto m^h\text{.}\) Then the induced homomorphisms \(H^i(G,M) \to H^i(G,M)\) are all identity maps.
  2. If \(H\) is a subgroup of \(G\text{,}\) \(M\) is a \(G\)-module, and \(M'\) is just \(M\) with all but the \(H\)-action forgotten, we get the restriction homomorphisms
    \begin{equation*} \Res\colon H^i(G, M) \to H^i(H, M)\text{.} \end{equation*}
    Another way to get the same map: use the adjunction homomorphism \(M \to \Ind^G_H M\) from Proposition 3.2.6 sending \(m\) to \(\sum_i m^{g_i} \otimes [g_i^{-1}]\text{,}\) where \(g_i\) runs over a set of right coset representatives of \(H\) in \(G\text{,}\) then apply Shapiro’s lemma (Lemma 3.2.3) to get
    \begin{equation*} H^i(G, M) \to H^i(G, \Ind^G_H M) \stackrel{\sim}{\to} H^i(H, M). \end{equation*}
  3. Let \(M\) be a \(G\)-module and consider the map \(\Ind^G_H M \to M\) taking \(m \otimes [g]\) to \(m^g\text{.}\) We then get maps \(H^i(G, \Ind^G_H M) \to H^i(G,M)\) which, together with the isomorphisms of Shapiro’s lemma (Lemma 3.2.3), give what are called the corestriction homomorphisms:
    \begin{equation*} \Cor\colon H^i(H, M) \stackrel{\sim}{\to} H^i(G, \Ind^G_H M) \to H^i(G, M). \end{equation*}
  4. The composition \(\Cor \circ \Res\) is induced by the homomorphism of \(G\)-modules \(M \to \Ind^G_H M \to M\) given by
    \begin{equation*} m \mapsto \sum_i m^{g_i} \otimes [g_i^{-1}] \to \sum_i m = [G:H]m. \end{equation*}
    Thus \(\Cor \circ \Res\) acts as multiplication by \([G:H]\) on each (co)homology group.
    Bonus consequence (hereafter excluding the case of \(H^0\)): if we take \(H\) to be the trivial group, then the group in the middle is isomorphic to \(H^i(H, M) = 0\text{.}\) So every cohomology group for \(G\) is killed by \(\#G\text{,}\) and in particular is a torsion group.
    In fact, if \(M\) is finitely generated as an abelian group, this means \(H^i(G, M)\) is always finite, because each of these will be finitely generated and torsion. Of course, this won’t happen in many of our favorite examples, e.g., \(H^i(\Gal(L/K), L^*)\) for \(L/K\) a finite Galois extension of fields.
  5. Let \(H\) be a normal subgroup of \(G\text{,}\) let \(\alpha\) be the surjection \(G \to G/H\text{,}\) and let \(\beta\) be the injection \(M^H \hookrightarrow M\text{.}\) Note that \(G/H\) acts on \(M^H\text{;}\) in this case, we get the inflation homomorphisms
    \begin{equation*} \Inf\colon H^i(G/H, M^H) \to H^i(G, M). \end{equation*}
    The inflation and restriction maps will interact in an interesting way; see Proposition 4.2.14.

Exercises Exercises

1.

Complete the proof of the correspondence between \(H^1(G,M)\) and principal homogeneous spaces (Definition 3.2.15).

2.

The set \(H^2(G,M)\) has the structure of an abelian group. Describe the corresponding structure on short exact sequences \(0 \to M \to E \to G \to 0\text{.}\) (A related concept in homological algebra is the Baer sum.)
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