I'm reading the Proposition 12.18 in the Gortz's Algebraic Geometry and stuck at some statement.
Let $k \subset k' \subset \Omega$ be a field extension where $\Omega$ is algebraically closed. Let $A$ be a finite $k$-local algebra with residue field $\kappa(x)$, so that it is an artinian : Atiyah-Macdonald, Exercise 8.3: Artinian iff finite k-algebra..
Let $X' := \operatorname{Spec}(A\otimes_k k').$ Note that since $\operatorname{dim}_{k^{'}}A\otimes_k k' = \operatorname{dim}_kA <\infty$, $A\otimes_k k'$ is also artinian and $X'$ is a finite set ( $\because$ Eisenbud, Commutative Algebra, Theorem 2.14). Let $X'=\{x_1', \cdots x_n' \}$.
Then
Q.1 ) $[\kappa(x) : k]_{sep}=\#\operatorname{Hom}_k(A, \Omega)$ ( $[\kappa(x) : k]_{sep}$ is the separable degree) ?
Q.2 ) $\# \operatorname{Hom}_{k'}(A\otimes_k k' , \Omega) = \Sigma_{x'\in X'}[\kappa(x') : k']_{sep}$ ?
These are true? Why? For the second question, my first attempt is as follows :
Let $B:= A\otimes_k k'$. Then since $B$ is artinian, the natural map
$$ B \to \prod_{\mathfrak{p}_{x_i'} \in \operatorname{Spec}B}B_{\mathfrak{p}_{x_i'}}$$
is an isomorphism. So,
$$ \# \operatorname{Hom}_{k'}(A\otimes_kk', \Omega) = \# \operatorname{Hom}_{k'} (\prod_{i=1}^{n}B_{\mathfrak{p}_{x_i'}} , \Omega) \overset{\mathrm{?}}{=} \Sigma_{i=1}^{n}\#\operatorname{Hom}_{k'}(B_{\mathfrak{p}_{x_i'}}, \Omega) = \Sigma_{i=1}^{n}\#\operatorname{Hom}_{k'}(\mathcal{O}_{X',x_i'} , \Omega) \overset{\mathrm{Q.1)}}{=} \Sigma_{i=1}^{n}[\kappa(x_i') : k']_{sep} $$
And is the second equality true?
This question originates from proof of the Proposition 12.18 in the Gortz's book (as a reference, I also upload image which includes definitions of notations which are appeared in the proposition 12.18) :
(Note that the first formula in the Proposition 12.18 is false : refer to the p.328 in https://www.algebraic-geometry.de/errata/1/ )
I think that I understand the proof of Proposition 12.18 (If needed, I will upload details.) except the underlined statements. Why the below underlined equalities are true?
Can anyone helps?
