# Proof of Prop 1.1 in Wiles' "Modular elliptic curves and Fermat's last theorem"

On p. 459 of "Modular elliptic curves and Fermat's last theorem", proof of Prop. 1.1, where it says "Since $$H^2(G,\mu_{p^r}) \rightarrow H^2(G,\mu_{p^s})$$ is injective for $$r \leq s$$...", is there any chance that that's a typo and he really means to say $$H^1$$ both times rather than $$H^2$$?

He uses it to conclude that the tensor product of one $$H^1$$ with a certain module is isomorphic to another $$H^1$$, so if the hypothesis really is with an $$H^2$$ both times then that presumably must involve exact sequences in some way. I'd appreciate any help in understanding the argument.

Well, although there is a typo (Wiles forgot to close his parenthesis, and wrote $$H^2(G,\mu_{p^r}\to H^2(G,\mu_{p^s})$$ in his proof), his claim is correct. Let, as ibid. $$F$$ be the finite extension of $$\mathbb{Q}_p$$ fixed by $$G$$, so that your arrow can be written $$H^2(F,\mu_{p^r})\to H^2(F,\mu_{p^s})$$. Consider the Kummer sequence (for any $$n\geq 1$$) $$1\longrightarrow \mu_{p^n}\longrightarrow \overline{F}^\times\overset{(\cdot)^{ p^n}}{\longrightarrow}\overline{F}^\times\longrightarrow 1.$$ Taking Galois cohomology and using Hilbert '90, which tells you $$H^1(F,\overline{F}^\times)=1$$, you find $$1\longrightarrow H^2(F,\mu_{p^n})\longrightarrow H^2(F,\overline{F}^\times)\overset{\cdot p^n}{\longrightarrow }H^2(F,\overline{F}^\times).$$ Now, local class field theory tells you that the Brauer group $$H^2(F,\overline{F}^\times)$$ is isomorphic to $$\mathbb{Q}/\mathbb{Z}$$, so the above sequence identifies $$H^2(F,\mu_{p^n})$$ with the group $$\mathbb{Z}/p^n$$, seen as the kernel of multiplication by $$p^n$$ on $$\mathbb{Q}/\mathbb{Z}$$. If now you apply this for $$r\leq s$$, you see that the arrow you were firstly interested in is the injection $$\mathbb{Z}/p^r\hookrightarrow \mathbb{Z}/p^s$$ or, if you prefer, the injection $$\Big(\ker(\cdot p^r)\hookrightarrow \ker(\cdot p^s)\Big)\subseteq \mathbb{Q}/\mathbb{Z}.$$
Added Jan 24th Concerning the second exact sequence (that involving $$H^1$$), there, there is a typo! What Wiles wanted to write was that the natural map $$\tag{1} H^1(G,\mu_{p^n})\otimes M\longrightarrow H^1(G,M(1))$$ is an isomorphism, but in his paper he made (again!) a mistake with parenthesis and ibid he replaces the first term by $$H^1(G,\mu_{p^n}\otimes M)$$. You can see (1) in the quoted paper by Diamond and, moreover, it is (1) which shows up later in Wiles' proof: indeed, he deduces from (1) something about another sequence where $$H^1(G,\mu_{p^n})\otimes M$$ is replaced, by Kummer theory, by $$((F^\times/(F^\times)^{p^n})\otimes M$$, which he could not do had he only some result about $$H^1(G,\mu_{p^n}\otimes M)$$.
To prove (1), observe that Wiles has managed to make the action of $$G$$ on $$M$$ trivial, and $$n$$ is such that $$p^nM=0$$. So, $$M$$ is a finite abelian $$p$$-group of exponent bounded by $$n$$, and isomorphic (as $$G$$-module!) to a finite number of copies of $$\mathbb{Z}/p^a\mathbb{Z}$$ for $$a\leq n$$. It thus suffices to prove (1) assuming $$M=\mathbb{Z}/p^a$$ for $$a\leq n$$. Consider, for any $$s\geq 0$$, the exact sequence $$0\longrightarrow \mu_{p^s}{\longrightarrow}\mu_{p^{s+a}}\longrightarrow \mu_{p^a}\longrightarrow 0.$$ By the first result on $$H^2$$ it gives rise to a surjective map $$H^1(G, \mu_{p^s}){\longrightarrow}H^1(G,\mu_{p^{s+a}})\longrightarrow H^1(G,\mu_{p^a})\longrightarrow 0$$ and, by taking projective limit over $$s$$, $$H^1(G, \mathbb{Z}_p(1))\overset{p^a}{\longrightarrow}H^1(G,\mathbb{Z}_p(1)\longrightarrow H^1(G,\mu_{p^a})\longrightarrow 0$$ which means $$H^1(G, \mathbb{Z}_p(1))/p^aH^1(G, \mathbb{Z}_p(1))\cong H^1(G,\mu_{p^a})$$. But this can be rewritten as $$H^1(G, \mathbb{Z}_p(1))\otimes\mathbb{Z}/p^a\cong H^1(G, \mathbb{Z}/p^a(1)).$$ This is (1) for the module $$\mathbb{Z}/p^a$$ and finished the proof.