My question refers to the proof of Proposition 2.4, p. 341 in Lang's Algebra. Here is the context:
Let $A$ be an integral domain, integrally closed in its field of quotients $K$ and let $L$ be a finite Galois extension of $K$ with group $G$. Let $p$ be a maximal ideal of $A$ and let $\beta$ be a maximal ideal of $B$ that lies above $p$, i.e. $A \cap \beta=p$. Let $G_{\beta}$ be the subgroup of $G$ consisting of those automorphisms $\sigma$ such that $\sigma \beta = \beta$. Let $L^{dec}$ be the fixed field of $G_{\beta}$ in $L$ and let $B^{dec}$ be the integral closure of $A$ in $L^{dec}$. We know that $\beta$ is the only prime ideal of $B$ lying over $Q=\beta \cap B^{dec}$ in $B^{dec}$.
In the proof of Proposition 2.4, it is argued that for any $\sigma$ not inside $G_{\beta}$ we have that $Q_{\sigma} = \sigma^{-1} \beta \cap B^{dec} \neq Q$. This is true because if that was not the case, $\sigma^{-1} \beta$ would also lie above $Q$ and this is a contradiction because it would imply that $\beta = \sigma \beta$. Then the chinese remainder theorem is invoked to show the existence of a $y \in B^{dec}$ so that given $x \in B^{dec}$ we have that $y = x (mod Q)$ and $y = 1 (mod Q_{\sigma})$ for all $\sigma$ not inside $G_{\beta}$. To use the chinese remainder theorem, we need at least the ideals to be distinct. In particular we want $Q_{\sigma} \neq Q_{\tau}$ for any $\sigma, \tau$ not inside $G_{\beta}$. Why is that true?
My concern: consider $\sigma, \tau$ be distinct elements of $G$, not inside $G_{\beta}$ but inside the same coset of $G_{\beta}$ in $G$. Then $\sigma \beta = \tau \beta$ and consequently $Q_{\sigma} = Q_{\tau}$.
Added:
This is how his proof continues. There exists such a $y$ as above and so $y=x (mod \, \beta), y=1 (mod \, \sigma^{-1} \beta)$ for each $\sigma$ not in $G_{\beta}$. The second congruence gives $\sigma y=1 (mod \, \beta)$ for all $\sigma$ not in $G_{\beta}$. Taking the norm of $y$ from $L^{dec}$ to $K$ gives $N^{L^{dec}}_K=x (mod \, \beta)$. This later congruence is also true $mod \, B^{dec}$ and so $mod \, Q$, from which he concludes that $A/p = B^{dec} / Q$ (what we want to prove), since the norm is in $A$.
Any insights? Thanks.