## augmentation ideal and $\rho$-isotopic spaces

These are two questions regarding Rubins paper – Global units and Ideal class groups in Inventiones 1987. The two questions are from section 3.

Let $$p$$ be a rational prime and $$K \subset E \subset F$$, where $$F$$ is an abelian extension of $$K$$ containing the Hilbert Class Field of $$K$$, with $$G=Gal(F/K)$$. Let $$\rho$$ be an irreducible $$\mathbb{Z}_p$$-representation of $$\Delta = Gal(F/E)$$. For any $$\mathbb{Z}[\Delta]$$-module $$M$$, write $$M^{\rho}$$ for $$\rho$$-eigenspace of $$\hat{M}= \lim M/p^n M$$. Explicitly, $$M^{\rho}= e_{\rho}\hat{M}$$, where $$e_{\rho}$$ is the $$\rho$$-idempotent. Let $$W$$ be a submodule of $$(O_F^*)^{\rho}$$ such that $$(O_F^*)^{\rho}$$ has no $$\mathbb{Z}_p$$ torsion. Let $$N=p^n$$.

Since $$\rho \ne 1$$, I have two questions:

1) Why is $$(\mathbb{Z}/N\mathbb{Z})[G]^{\rho}$$ contained in the augmentation ideal of $$(\mathbb{Z}/N\mathbb{Z})[G]$$? I couldnt understand the action of $$\rho$$ on a $$(\mathbb{Z}/N\mathbb{Z})[G]$$-module $$M$$.

2) Let $$V=W/W^N$$. Why is $$O_K^* \cap V = 0$$?