In Kato's $p$-adic Hodge Theory and Values of Zeta Functions of Modular Forms, Kato proves in 13.8 that the first Iwasawa cohomology: $$\mathbf{H}^1(T):=\varprojlim_n H^1(\mathbb{Z}[\zeta_{p^n}, 1/p],T)$$ is free of rank one under the hypothesis that the residual Galois representation: $$T/\mathcal{m}_\lambda T$$ 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. I have a number of questions about this.
- Is the integer $r$ in this statement independent of the choice of $x,y$?
- 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)
- Could Kato sharpen his hypothesis from "$T/\mathcal{m}_\lambda T$ is irreducible" to "the $r$-th Tate twist of the trivial character does not occur in $T/\mathcal{m}_\lambda T$", or is there something more technical to the proof that I'm missing?