On Kato's proof that Iwasawa Cohomology is free

Question

In Kato's $p$-adic Hodge Theory and Values of Zeta Functions of Modular Forms, he proves in 13.8 that the first Iwasawa cohomology: $$\mathbf{H}^1(T):=\varprojlim_nH^1(\mathbb{Z}[ζpn,1/p],T)$$ is free of rank one under the hypothesis that the residual Galois representation, $T/\mathfrak{m}_\lambda T$ is irreducible. In particular Kato uses the fact that given a maximal ideal $(x,y)$ in $\Lambda$, $$\Lambda/(x,y)\cong\mathcal{O}_\lambda/\mathfrak{m}_\lambda(r)$$ where $(r)$ denotes some Tate twist.

1. Is the integer $r$ in this statement independent of $x,y$?
2. If it is independent, then is it known which integer r occurs here? (it is unclear to me how anything but the trivial representation could occur here)
3. Could Kato sharpen his hypothesis from "$T/\mathfrak{m}_\lambda T$ is irreducible" to "the r-th Tate twist of the trivial character does not occur in T/mλT", or is there something more technical to the proof that I'm missing? (This is a cross-post from math stack exchange. Apologies if this question isn't formatted as well as it could be.)

Answer 1

(1) No, it depends on the maximal ideal (maximal ideals of $\Lambda$ biject with mod $p$ characters of $\Gamma$).

(2) no longer makes sense.

(3) It would suffice (for this particular step of Kato's proof) to assume that $H^0(\mathbb{Q}, T / \mathfrak{m}_\lambda T (r)) = 0$ for $0 \le r < p-1$, or equivalently that $H^0(\mathbb{Q}(\zeta_p), T / \mathfrak{m}_\lambda T) = 0$.