# Decomposition and valuation rings

I am reading Algebraic Number Theory by A. Fröhlich, M. J. Taylor, it first introduced the theorem:

$$(K,u)$$ 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 theorem, 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: • Please note that "theory" and "theorem" are different words. Aug 23, 2015 at 14:39
• @ZevChonoles Thanks, I am not naive English speaker and sometimes misuse.
– CYC
Aug 23, 2015 at 14:44

I would say from context that $\mathfrak{o}_L$ and $\mathfrak{o}_K=\mathfrak{o}$ are the rings of integers in $L$ and $K$ respectively (thus, the intersection of all the valuation rings for the different non-archimedean valuations) and $\mathfrak{o}_{L_{w_i}}$ is the valuation ring for $w_i$.
The notation $\mathfrak{o}_K$ (hence also $\mathfrak{o}_L$) is explained on Page 2 of the book:
The most important such subring is $\mathfrak{o}_K$, the set of all algebraic integers in $K$: later we shall show that $\mathfrak{o}_K$ is indeed a ring.
The notation $\mathfrak{o}_{L_{w_i}}$ should be clear from the context, since it is analogous to $\mathfrak{o}_{K_u}$.