I am reading Algebraic Number Theory by A. Fröhlich, M. J. Taylor, it first introduced the theory:
$(K,u)$ be a field and its absolute value, $(K_u,\bar u)$ be its completion and absolute value extension, $L/K$ a finite separable field extension.
Then the $K_u$-algebra $K_u\otimes_K L \approx \prod_i (K_u\otimes_K L)\eta_i \approx \prod_{w_i} L_{w_{i}}$.
( $\eta_i$'s are idempotents of $K_u\otimes_K L$, $L_{w_i}$'s are finite separable field extension of $K_u$ where $w_i$ runs through all absolute values of L which extend u.)
I have no problem with this theory, my question is the next proposition:
With the same notation above, let $\mathfrak p$ be a prime ideal of a Dedekind domain $\mathfrak o$ with field of fraction $K$, and if $u$ is a discrete absolute value of $K$ corresponding to $\mathfrak p$, then, writing $\mathfrak o_{K_u}$ for the valuation ting in $K_u$($\mathfrak o_{K_u}=\{x;\bar u(x)\leqslant 1\}$), we have $\mathfrak o_{K_u}\otimes_{\mathfrak o_K} \mathfrak o_L \approx \prod_i (\mathfrak o_{K_u}\otimes_{\mathfrak o_K} \mathfrak o_L)\eta_i \approx \prod_{w_i} \mathfrak o_{L_{w_i}}$
The book doesn't specify what $\mathfrak o_L, \mathfrak o_K, \mathfrak o_{L_{w_i}}$ are, but with the convention notations in the book, it might mean valuation rings in $L, K, L_{w_i}$ respectively.
However, since $L$ has multiple distinct absolute values ($w_i$'s), it's not clear $\mathfrak o_L$ corresponds to which absolute value of $L$, hence I have no idea what this property is talking about. What are $\mathfrak o_L, \mathfrak o_{w_i}$?
This is its proof which may help understanding:
(1.3 in the proof is here)
(u-norm definition)
