I have the following problem.
Let $L/K$ be a finite galois extension of local fields with Galois group $G$. For nontrivial $g\in G$ define Lefschetz number $i_{L/K}(g):= \min\limits_{x\in \mathcal{O}_L}v_L(g(x)-x)$, where $\mathcal{O}_L$ is a valuation ring in $L$ and $v_L$ is normalized valuation.
My problem is following. Why $i_{L/K}(g)$ is equal to the intersection multiplicity of diagonal and graph of $g$ in $Spec\: \mathcal{O}_L \times_{Spec\: \mathcal{O}_K} Spec\: \mathcal{O}_L$?
Since $K$ and $L$ are local, we have $\mathcal{O}_L = \mathcal{O}_K[\pi]$, and we can simplify definition of $i_{L/K} $: $i_{L/K}(g)= v_L(g(\pi)-\pi)$. Also $\pi$ is a uniformizer at special point in $Spec\: \mathcal{O}_L$, so the definition of $i_{L/K}(g)$ is similar to the definition of multiplicity of plain algebraic curves, but I actually can’t understand why it is the same. Any help would be appreciated