Finite-fibre and pointwise characterizations of quasi-finiteness

Fix $x\in X$ and $s=f(x)$. If (1) holds, take compatible affine neighbourhoods $U=\mathrm{Spec}B$ and $V=\mathrm{Spec}A$ from [F1], with $f(U)\subseteq V$, and let $\mathfrak{q}$ and $\mathfrak{p}=\mathfrak{q}\cap A$ correspond to $x$ and $s$. The local ring of $X_s$ at $x$ is $B_{\mathfrak{q}}\mathrm{/}\mathfrak{p}{B}_{\mathfrak{q}}$ by [F1] and [F4]. The witnessing map is finite type, so [F2] says this local fibre algebra is finite-dimensional over $\kappa (s)$. For the reverse implication, choose a compatible affine pair witnessing that $f$ is locally of finite type, as provided by [F3]. Its fibre chart $U_s=U{\times }_V\mathrm{Spec}\kappa (s)$ is open in $X_s$, so isolatedness restricts to $U_s$. The same local-ring identification and the same criterion then apply. Stacks Commutative Algebra tag 00PK, whose proof reduces to tag 00PJ proves that finite-dimensionality of this local fibre algebra is equivalent to $x$ being isolated in $X_s$. Its equivalent residue-field condition, and directly the quotient map from the finite-dimensional local algebra to $\kappa (x)=B_{\mathfrak{q}}\mathrm{/}\mathfrak{q}{B}_{\mathfrak{q}}$, give that $\kappa (x)\mathrm{/}\kappa (s)$ is finite. Thus (1) and (2) are equivalent pointwise; the same scheme-level isolated-point and residue-field arguments are recorded in Stacks Morphisms tag 01TC.

Suppose the fibre morphism $X_s\to \mathrm{Spec}\kappa (s)$ is finite. By [F5], $X_s=\mathrm{Spec}C$ with $C$ finite-dimensional over $\kappa (s)$. Such a ring is Artinian and is a finite product of local Artinian rings. Its spectrum is therefore a finite discrete space, and each residue field is a quotient of a finite-dimensional $\kappa (s)$-vector space. Hence every point of this fibre is isolated and has finite residue extension. The argument includes nilpotents; for example, $\kappa (s)[ϵ]\mathrm{/}({ϵ}^2)$ has one isolated point with residue field $\kappa (s)$. If $X_s$ is empty then $C=0$ and the pointwise conclusions are vacuous. Thus (3) implies (2).

Assume (1). By step 1.1 every point of each fibre is isolated, so each fibre is zero-dimensional. Fix $s\in S$. The base change $X_s\to \mathrm{Spec}\kappa (s)$ is finite type by [F6], and it is quasi-compact because $f$ is finite type by [F3] and quasi-compactness survives base change by [F6]. Let $W=\mathrm{Spec}D$ be any affine open of $X_s$. Its finite-type $\kappa (s)$-algebra $D$ has dimension zero. As in the complete proof at Stacks Varieties tag 06LH, Noether normalization makes $D$ finite-dimensional over $\kappa (s)$; therefore $D$ is Artinian and decomposes into finitely many local Artinian factors. Each such factor gives an open singleton in $W$, hence in $X_s$. These singleton opens cover $X_s$. Quasi-compactness gives a finite subcover, so $X_s$ is a finite disjoint union of spectra of finite-dimensional local Artinian $\kappa (s)$-algebras. Their finite product is a finite-dimensional coordinate algebra, and [F5] says the resulting morphism to $\mathrm{Spec}\kappa (s)$ is finite. The empty fibre is $\mathrm{Spec}(0)$ and is finite as well. This proves (1) implies (3), including the one-point case $\mathrm{Spec}\kappa (s)$.

Step 1.1 proves (1)$⟺$(2), step 2.1 proves (1)$⟹$(3), and step 1.2 proves (3)$⟹$(2). These three implications give all directions among the stated conditions. The proof makes no AC assumption: for each fixed fibre its quasi-compactness supplies a finite subcover, and no family of choices is assembled.