Finite-fibre and pointwise characterizations of quasi-finiteness

We first record a choice-free existence fact used only for finite-type field algebras. A nonzero finite-type $k$-algebra $T$ has a maximal ideal. By [F8] it is module-finite over an injected polynomial ring $R=k[z_1,\dots ,z_d]$. Localize at the explicit maximal ideal $\mathfrak{m}=(z_1,\dots ,z_d)$. The injection $R_{\mathfrak{m}}\hookrightarrow T_{\mathfrak{m}}$ shows the latter is nonzero and finite as an $R_{\mathfrak{m}}$-module. If $\mathfrak{m}{T}_{\mathfrak{m}}=T_{\mathfrak{m}}$, the determinant trick in [F8] gives $(1-a)T_{\mathfrak{m}}=0$ with $a\in \mathfrak{m}{R}_{\mathfrak{m}}$; $1-a$ is a unit in this localization, a contradiction. Hence $T\mathrm{/}\mathfrak{m}T$ is a nonzero finite-dimensional $k$-algebra. Among its proper ideals choose one of maximum vector-space dimension; such a dimension is a largest member of a nonempty finite subset of the natural numbers. That ideal is maximal, and its inverse image in $T$ is maximal. Only finitely many choices and finite-dimensional linear algebra occur.

For a finite-type $k$-algebra $A$, its nilradical is the intersection of its prime ideals without requiring the general prime-ideal theorem. Indeed, if $a$ is not nilpotent, the nonzero finite-type algebra $A[a^{-1}]$ has a maximal ideal by step 1.1; its contraction is prime and avoids $a$. The reverse inclusion, that every prime contains each nilpotent, follows by taking a power. By [F7] the nilradical is finitely generated, say by $a_i$ with $a_i^{n_i}=0$, and is nilpotent: every monomial of degree ${\sum }_i{n}_i$ in the $a_i$ contains a vanishing power. Thus if $A$ has just one prime, its maximal ideal is nilpotent. The zero ring is excluded only while a prime is being considered.

Let $D$ be a finite-type $k$-algebra and $\mathfrak{q}\in \mathrm{Spec}D$. Suppose $D_{\mathfrak{q}}$ is finite-dimensional over $k$. Its residue field is finite-dimensional. The domain $D\mathrm{/}\mathfrak{q}$ injects into that field, hence is itself finite-dimensional; multiplication by a nonzero element is an injective endomorphism of a finite-dimensional space and is surjective, so this domain is a field. Thus $\mathfrak{q}$ is maximal. Every prime quotient of the finite-dimensional algebra $D_{\mathfrak{q}}$ is similarly a field, so its only prime is its unique maximal ideal $\mathfrak{q}{D}_{\mathfrak{q}}$. By step 2.1 this ideal is nilpotent. Since $D$ is Noetherian by [F7], choose generators of ${\mathfrak{q}}^N$ for a power that becomes zero in $D_{\mathfrak{q}}$. Each is annihilated by some element outside $\mathfrak{q}$; their finite product $s\mathrm{\notin }\mathfrak{q}$ kills all of ${\mathfrak{q}}^N$. Every prime of $D_s$ contains $\mathfrak{q}{D}_s$, since its $N$th power vanishes, and maximality of $\mathfrak{q}$ leaves precisely that one prime. Consequently $D(s)=\{\mathfrak{q}\}$ in $\mathrm{Spec}D$. This proves isolatedness and finite residue field from finite-dimensionality of the local algebra.

Conversely suppose $\mathfrak{q}$ is isolated and its residue field is finite over $k$. Choose a principal neighbourhood $D(s)=\{\mathfrak{q}\}$. The finite-type algebra $A=D_s$ has just that prime. It is local: if an element outside that prime were not a unit, its nonzero finite-type quotient would have a maximal ideal by step 1.1, yielding a second prime. Thus $A=D_{\mathfrak{q}}$. By step 2.1 its maximal ideal $\mathfrak{n}$ is nilpotent. All powers of $\mathfrak{n}$ are finitely generated by [F7], so the quotients ${\mathfrak{n}}^i\mathrm{/}{\mathfrak{n}}^{i+1}$ are finite-dimensional over $A\mathrm{/}\mathfrak{n}=\kappa (\mathfrak{q})$: the images of finite ideal generators span them. This residue field is finite over $k$ by hypothesis. The finite filtration by the powers ending at ${\mathfrak{n}}^N=0$ proves that $A=D_{\mathfrak{q}}$ is finite-dimensional over $k$. This proves the reverse local criterion, including nilpotents.

For $x\in X_s$, a compatible finite-type affine chart has fibre algebra $D=B{\otimes }_A\kappa (s)$ and its local ring at $x$ is $B_{\mathfrak{q}}\mathrm{/}\mathfrak{p}{B}_{\mathfrak{q}}$ by [F1], [F2] and [F4]. Steps 3.1–3.2 identify finite-dimensionality of this ring with isolatedness and finite residue extension in that affine fibre chart, hence in $X_s$. This is an open-local assertion. Definitions [F1]–[F3] therefore give (1)$⟺$(2). If these hold, every point of $X_s$ has a singleton principal affine neighbourhood whose coordinate ring is its finite-dimensional local ring, by step 3.2. Those opens cover the fibre. Its quasi-compactness follows from finite type and [F6], so finitely many suffice. They are disjoint singleton opens, and their finite disjoint union is the spectrum of the finite product of their finite-dimensional algebras. Thus $X_s$ is affine finite-dimensional over $\kappa (s)$, proving (3) by [F5]. No separation hypothesis is used in this disjoint open cover.

Conversely assume (3). Write $X_s=\mathrm{Spec}D$ with $D$ finite-dimensional over $\kappa (s)$ by [F5]. Every localization $D_{\mathfrak{q}}$ is still finite-dimensional. To see this explicitly, for each inverted element $b\mathrm{\notin }\mathfrak{q}$ take a polynomial relation $b^{r}g(b)=0$ with $g(0)\mathrm{\ne }0$; one exists by finite-dimensional linear dependence and removal of the largest power of the variable. After inverting $b$ one has $g(b)=0$, so $b^{-1}$ is a polynomial in $b$ using the nonzero constant coefficient. Hence the localization map $D\to D_{\mathfrak{q}}$ is surjective and its target is finite-dimensional. Step 3.1 gives isolatedness and finite residue extension for each point, proving (2). An empty fibre is $\mathrm{Spec}(0)$ and is finite; its pointwise conditions are vacuous.

Step 4.1 proves (1)$⟺$(2) and (1)$⟹$(3); step 4.2 proves (3)$⟹$(2). These give all claimed equivalences. The Noetherian and normalization suppliers in [F7]–[F8] use only their choice-free finite-generation implications and finite selections; step 1.1 explicitly replaces a general maximal-ideal existence appeal. No choice principle or external Stacks proof is an input.