Quasi-finite morphisms of schemes

A morphism of schemes $f:X\to S$ is quasi-finite if it is of finite type (Locally finite type and finite type morphisms) and, for every point $x\in X$, there are affine neighbourhoods $U=\mathrm{Spec}(B)$ of $x$ and $V=\mathrm{Spec}(A)$ of $f(x)$ such that $f(U)\subseteq V$ and the induced finite-type ring map $A\to B$ is quasi-finite at the prime $\mathfrak{q}\subset B$ corresponding to $x$, in the sense of Quasi-finiteness at a prime of a finite-type algebra. If $\mathfrak{p}=\mathfrak{q}\cap A$, the local algebra in that condition is $B_{\mathfrak{q}}\mathrm{/}\mathfrak{p}{B}_{\mathfrak{q}}\mathrm{.}$ It is the local ring of the scheme-theoretic fibre $X_{f(x)}$ at $x$ (Scheme-theoretic fibre).

The empty source satisfies the pointwise condition vacuously. No separatedness condition is part of the definition; any theorem that also needs separatedness states it separately.