Question is simple.
Let $\varphi : R \to A $ be a finite type ring map with $\mathfrak{p}$ a prime ideal of $A$. Let $\mathfrak{q} := \varphi^{-1}(\mathfrak{p})$ and $\psi : R_{\mathfrak{q}} \to A_{\mathfrak{p}}$ be the induced local homomorphism. Then the induced extension of residue fields $R_{\mathfrak{q}}/{\mathfrak{q}}R_{\mathfrak{q}} \to A_{\mathfrak{p}}/{{\mathfrak{p}}}A_{\mathfrak{p}}$ is finitely generated?
This question originates from trial to show that "If $ f: X\to Y$ is a morphism of finite type with $x\in X$ and $y:=f(x)$, then $\kappa(x)$ is finitely generated field extension of $\kappa(y)$."
Since $f$ is of finite type at $x\in X$, there exists affine neighborhoods $x\in \operatorname{Spec}A =U \subseteq X$, $y:=f(x)\in \operatorname{spec}R=V \subseteq Y$ with $f(U) \subseteq V$ such that the ring map
$$\varphi := (f|_{U}^{V})^{\flat}_V : R \to A $$
is of finite type. $\varphi$ induces morphism of affine schemes $ (f|_{U}^V, (f|_{U}^V)^{\flat})$.
And the induced local homomorphism $\psi := (f|_U^V)^{\sharp}_x :\mathcal{O}_{Y,y}=R_{\mathfrak{p}_y = \varphi^{-1}(\mathfrak{p}_x)} \to \mathcal{O}_{X,x}= A_{\mathfrak{p}_x} $
is induced from $\varphi$.
So, if our question is true, then the induced map between residue fields $\kappa(y) \to \kappa(x)$ is finitely generated.
Can anyone helps?
Best Answer
Localization commutes with taking quotients, so we may view the extension of residue fields as the extension of fraction fields of induced by $R/\mathfrak{q} \to A/\mathfrak{p}$. If $A$ is finite type over $R$, then $A/\mathfrak{p}$ is finite type over $R/\mathfrak{q}$ (generators for $A$ over $R$ descend to generators for $A/\mathfrak{p}$ over $R/\mathfrak{q}$). Thus the finitely many generators generate $\mathrm{Frac} (A/\mathfrak{p}) \simeq A_\mathfrak{p}/\mathfrak{p} A_{\mathfrak{p}}$ as a field over $\mathrm{Frac} (R/\mathfrak{q}) \simeq R_\mathfrak{q}/\mathfrak{q} R_{\mathfrak{q}}$. (When we say "finitely generated as a field," we allow elements to be inverted; this is different than being finitely generated as an algebra.)