# Traces in finite extensions of integrally closed domains

$$\def\fp{\mathfrak{p}}\def\fq{\mathfrak{q}}$$I'm looking for a reference for the following commutative algebra fact.

Let $$A$$ be an integrally closed integral domain, with field of fractions $$K$$. Let $$L$$ be a finite extension of $$K$$ and let $$B$$ be the integral closure of $$A$$ in $$L$$. Let $$\fp$$ be a prime of $$A$$ and let $$\fq_1$$, $$\fq_2$$, ..., $$\fq_r$$ be the primes of $$B$$ containing $$\fp B$$. Let $$k = \mathrm{Frac}(A/\fp)$$ and $$\ell_i = \mathrm{Frac}(B/\fq_i)$$.

Then there are positive integers $$e_1$$, $$e_2$$, ..., $$e_k$$ such that, for $$\theta \in B$$, $$\mathrm{Tr}_{L/K} \theta \equiv \sum e_i \mathrm{Tr}_{\ell_i/k} \theta \bmod \fp.$$

Here we have to be a little careful as $$\dim_k \ell_i$$ can be infinite, in this case we will take $$e_i=0$$ so there is no contribution.

I am fairly sure I can prove this. Specifically, the recipe is as follows: If $$L/K$$ is not separable, take all $$e_i=0$$. Otherwise, let $$M$$ be the Galois closure of $$L$$ over $$K$$, with Galois group $$G$$ and let $$H$$ be the stabilizer of $$L$$. Let $$C$$ be the integral closure of $$A$$ in $$M$$. Let $$I_i \subset G$$ be the elements of the Galois group which preserve $$\fq_i C$$ and act trivially on $$\ell_i$$ inside $$C/\fq_i C$$. Then we take $$e_i = [I_i:H]$$. The proof is just carefully working through the formula $$\mathrm{Tr}_{L/K} \theta = \sum_{g \in G/H} g \theta$$, and noticing that, whenever something could go wrong because of inseparability, the trace is zero anyway.

But I can only find sources which want to discuss the Dedekind domain case. Any suggestions?

Of course, maybe I am wrong!