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.
Best Answer
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.