# Cosets of ramified Witt vectors

I am trying to solve exercise 1.5.1. of Peter Schneider's book Galois Representations and (Phi, Gamma)-Modules.

Specifically, let $L$ be a finite extension of the $p$-adic numbers $\mathbb{Q}_p$ and $\mathcal{O}_L$ its ring of integers with residue class field $k$ of cardinality $q=p^f$. Let $B$ be a perfect $k$-algebra (which, by Schneider's definition, means that the map $B \rightarrow B, b\mapsto b^q$ is bijective). Let $W(B)_L$ then be the ring of ramified Witt vectors over $L$. Set-theoretically, $W(B)_L$ is just $B^{\mathbb{N}}$, but the addition and multiplication are given in a rather complicated manner.

What I want to show is that, given $a=(a_0,a_1,...)\in W(B)_L$ and an ideal $\mathfrak{a}\subseteq B$, we have the following equality of subsets of $W(B)_L$: \begin{align} a + \{(b_0,...)\in W(B)_L: b_0,...,b_{m-1}\in \mathfrak{a} \} = \{(b_0,...)\in W(B)_L: b_i \equiv a_i \mod \mathfrak{a} \text{ for all } 0\leq i \leq m-1 \}. \end{align}

I sincerely hope that someone who is familiar with Witt vectors can shed some light on the matter.

• I may be way out of line here, but it looks as if it should fall out from the fact that multiplication and addition are given by $\mathscr O_L$-polynomials in the entries of the W-vectors. – Lubin May 17 '17 at 18:27