189-596A: Class Field Theory

Assignment 3. Due: Tuesday, October 30.

1. Let $K$ be a local field with uniformiser $\pi$, let $F$ be the Lubin-Tate formal group attached to $\pi$, and let $L:= K(F[\pi^n])$ be the field generated by the $\pi^n$-torsion on this formal group. Check that the group of norms from $L^\times$ to $K^\times$ is equal to $\pi^{\mathbb Z} (1+\pi^n\mathcal O_K)^\times$. (I claimed that we had essentially done this in class. The goal of this exercise is for you to spell out the details in the proof of this important fact.)

2. Use Lubin-Tate theory to write down an abelian extension of $K$ whose norm subgroup is $\pi^{n\mathbb Z} (1+\pi^m\mathcal O_K)^\times$, for $m,n\ge 1$.

3. (Taken from Kedlaya, (5), p. 34). Let $M$ be a $\mathbb Z[G]$-module, and let $0 \rightarrow M \rightarrow M_0 \rightarrow M_1 \rightarrow \cdots $ be an exact sequence of $\mathbb Z[G]$ modules in which the modules $M_i$ are free over $\mathbb Z[G]$. Show that the $i$-th cohomology group of the complex $$0 \rightarrow M_0^G \rightarrow M_1^G \rightarrow \cdots $$ is equal to $H^i(G,M)$. (Hint: use a dimension shifting argument, starting with the cohomology sequence attached to the short exact sequence

$ 0 \rightarrow M \rightarrow M_0 \rightarrow M_0/M \rightarrow 0$
using the fact that
$ 0 \rightarrow M_0/M \rightarrow M_1 \rightarrow M_2 \rightarrow \cdots$
is also exact.

4. Let $K$ be a field of characteristic $p>0$ and let $\bar K$ denote its seperable closure. Write $G_K$ for the Galois group of $\bar K/K$, endowed with its profinite (Krull) topology.

(a) Show that there is an exact sequence of $G_K$-modules $$ 0 \rightarrow (\mathbb Z/p\mathbb Z) \rightarrow \bar K \rightarrow \bar K \rightarrow 0,$$ where the penultimate map is given by $x\mapsto x^p-x$.

(b) By taking the (continuous) $G_K$-cohomology of this ecaxt sequence and invoking the additive counterpart of Hilbert's theorem 90, show that every cyclic extension of $K$ of degree $p$ is the splitting field of a polynomial of the form $x^p-x-a$ for a suitable $a\in K$.

5. Let $G$ be a finite group and let $A$ be a finite abelian group equipped with an action of $G$. Recall that we explained in class how a central extension $$ 0 \rightarrow A \rightarrow E \rightarrow G \rightarrow 0$$ gives rise to a two-cocycle $\varphi_E \in Z^2(G,A)$ which is well-defined up to elements of $B^2(G,A)$.

(a) Show that if two extensions $E_1$ and $E_2$ are isomorphic, i.e., if there is a commutative diagram

$ \begin{array}{rcccccc} 0 \rightarrow & A & \rightarrow & E_1 & \rightarrow & G & \rightarrow 0 \\ & \downarrow & & \downarrow & &\downarrow & \\ 0 \rightarrow & A & \rightarrow & E_2 & \rightarrow & G & \rightarrow 0 \end{array} $
in which the horidontal sequences are exact and the left and rightmost vertical maps are the identity, then the cocycles $\varphi_{E_1}$ and $\varphi_{E_2}$ are co-homologous (i.e., differ by a $2$-coboundary).

(b) Show conversely that if the classes of $\varphi_{E_1}$ and $\varphi_{E_2}$ in $H^2(G,A)$ are equal, then $E_1$ and $E_2$ are isomorphic extensions.

6. Given a class $\varphi\in H^2(G,A)$, construct an extension $E$ of $G$ by $A$ which satisfies $\varphi_E = \varphi$. (With notations as in the previous question.) Conclude that the assignment $E \mapsto \varphi_E$ is an isomorphism between ${\rm Ext}^1(G,A)$ and $H^2(G,A)$.

7. Let $A=G=\mathbb Z/p\mathbb Z$ with trivial action of $G$ on $A$. Compute $H^2(G,A)$ and give a description of the different group extensions of $G$ by $A$.

8. Let $G=S_n$ be the symmetric group on $n$ letters, and let $M=\mathbb Z^n$ be the abelian group on which $G$ acts by permutation of the standard basis.

(a) Show that $H^p(S_n,M) = H^p(S_{n-1},\mathbb Z)$ for all $p$, where the action of $S_{n-1}$ on $\mathbb Z$ is the trivial one.

(b) Compute the latter cohomology groups for $p=0,1,2$. (Hint: use the exact sequence $0 \rightarrow \mathbb Z \rightarrow \mathbb Q \rightarrow \mathbb Q/\mathbb Z \rightarrow 0$.)