A zero-dimensional projective scheme has finitely many closed points with finite-dimensional local rings

Each $A_i$ is generated as a $k$-algebra by the finitely many ratios $x_j\mathrm{/}{x}_i$, $j\mathrm{\ne }i$, computed in $S_{x_i}$: an element of $A_i$ has the form $s\mathrm{/}{x}_i^e$ with $s\in S$ homogeneous of degree $e$, the degree-$e$ piece of $S$ is spanned by the images of the monomials $x_0^{a_0}\cdots x_n^{a_n}$ with $a_0+\cdots +a_n=e$, and such a monomial satisfies $x_0^{a_0}\cdots x_n^{a_n}\mathrm{/}{x}_i^e={\prod }_{j\mathrm{\ne }i}(x_j\mathrm{/}{x}_i{)}^{a_j}$. Hence each $A_i$ is a finitely generated $k$-algebra, and $A_i=0$ exactly when the chart $D_{+}(x_i)$ is empty; by [L1] the finitely many nonempty charts cover $X$.

If $C$ is a finite-dimensional $k$-algebra, every $C$-submodule of $C$ is a $k$-subspace, and a strict inclusion of $C$-submodules strictly raises $k$-dimension. Starting with $C_0=C$, if $C_j\mathrm{\ne }0$, choose a proper $C$-submodule $C_{j+1}⊊C_j$ of largest possible $k$-dimension; $0$ is one candidate, and the possible dimensions lie in the finite set $\{0,\dots ,{\dim }_k{C}_j-1\}$. No submodule lies strictly between $C_{j+1}$ and $C_j$, since it would have larger dimension, so $C_j\mathrm{/}{C}_{j+1}$ is simple. The dimensions strictly decrease, hence the process reaches $0$ in at most ${\dim }_{k}C$ steps and gives a composition series of $C$-modules. Thus ${\mathrm{\ell }}_C(C)\le {\dim }_{k}C$. A flag of arbitrary $k$-basis spans would not suffice, because those spans need not be $C$-submodules.

If $A_i\mathrm{\ne }0$, then by [L4] there are algebraically independent $z_1,\dots ,z_d\in A_i$ such that $A_i$ is module-finite over $B=k[z_1,\dots ,z_d]$. The inclusion $B\subseteq A_i$ is then integral by [L5] and has zero kernel, so the kernel hypothesis of lying over is satisfied for $(0)$. If $d\ge 1$, then $(0)⊊(z_1)$ is a strict chain of primes of $B$ by [L7], lying over [L6] gives a prime ${\mathfrak{q}}_0$ of $A_i$ contracting to $(0)$, and the chain-lifting part of [L6] produces a prime ${\mathfrak{q}}_1\supseteq {\mathfrak{q}}_0$ contracting to $(z_1)$; since the two contractions differ, ${\mathfrak{q}}_0\mathrm{\ne }{\mathfrak{q}}_1$, so $A_i$ contains a strict chain of two primes, contradicting $\dim A_i=0$. Hence $d=0$, so $B=k$ and $A_i$ is a finite-dimensional $k$-vector space. In particular $A_i$ is Artinian by [L8]: a strictly descending chain of ideals of $A_i$ is a strictly descending chain of $k$-subspaces, and every strict inclusion strictly lowers the $k$-dimension, so no infinite strictly descending chain exists.

Let $A_i\mathrm{\ne }0$. By 2.1 it is Artinian, and its primes are maximal: a prime $\mathfrak{p}$ is contained in some maximal ideal $\mathfrak{m}$ by [L8], and $\mathfrak{p}⊊\mathfrak{m}$ would be a strict chain of two primes, contradicting $\dim A_i=0$. There are therefore only finitely many primes, they are the maximal ideals ${\mathfrak{m}}_{i,1},\dots ,{\mathfrak{m}}_{i,r_i}$ of $A_i$, and the structure theorem [L8] gives an isomorphism $A_i\cong {\prod }_j(A_i{)}_{{\mathfrak{m}}_{i,j}}$, under which the factor $(A_i{)}_{{\mathfrak{m}}_{i,j}}$ is a quotient of the finite-dimensional $k$-algebra $A_i$, hence finite-dimensional, local as a localization at a maximal ideal, and has nilpotent maximal ideal because $A_i\cong {\prod }_j{A}_i\mathrm{/}{\mathfrak{m}}_{i,j}^{n_j}$. The points of the chart $\mathrm{Spec}(A_i)$ are exactly these maximal ideals by [L1].

$X$ has finitely many points, all closed, whence its underlying space is finite and discrete. The charts are finitely many and each chart has the finitely many points ${\mathfrak{m}}_{i,1},\dots ,{\mathfrak{m}}_{i,r_i}$ of 3.1, so $X$ is finite. A subset $Z\subseteq X$ is closed exactly when every trace $Z\cap D_{+}(x_i)$ is closed in $D_{+}(x_i)$, because the charts are an open cover; for a point $x\in X$ the trace is empty whenever $x\mathrm{\notin }{D}_{+}(x_i)$, and otherwise it is the singleton $\{x\}\subseteq \mathrm{Spec}(A_i)$, which is closed because $x$ corresponds to a maximal ideal of $A_i$ by 3.1. Hence every point of $X$ is closed, and in a finite space with all points closed every subset is a finite union of closed points, so the space is discrete.

Let $x\in X$ and let $D_{+}(x_i)$ be any standard chart containing it. By 3.1 the point corresponds to a maximal ideal $\mathfrak{m}$ of the chart ring $A_i$, and by [L1] we have ${\mathcal{O}}_{X,x}\cong (A_i{)}_{\mathfrak{m}}$. By 3.1 this factor is a finite-dimensional local $k$-algebra with nilpotent maximal ideal and is a quotient of $A_i$; its residue field is $A_i\mathrm{/}\mathfrak{m}$, a quotient of the finite-dimensional $k$-algebra $A_i$, hence finite-dimensional over $k$; and it has finite length as a module over itself by 1.2. The same description holds for every chart containing $x$, and different charts give isomorphic local rings by the overlap statement in [L1].

Let $A_i\mathrm{\ne }0$. With the notation of step 3.1, write $A_i\cong {\prod }_j{B}_j$ where $B_j=(A_i{)}_{{\mathfrak{m}}_{i,j}}$. If $e_j$ is the coordinate idempotent of this product, a prime contains all but exactly one $e_j$: two omitted idempotents would have product zero, contrary to primality, and all cannot belong to a proper ideal because their sum is $1$. Thus every prime comes from one factor $B_j$. The maximal ideal of each local factor is nilpotent by step 3.1, so every prime contains it and must equal it; each factor has exactly one prime. Hence $\mathrm{Spec}(A_i)$ is the disjoint union of the spectra of the factors. By [L1] and step 3.1 each $B_j$ is the local ring ${\mathcal{O}}_{X,x}$ at the corresponding point $x$, so $\mathrm{Spec}(A_i)={⨆}_{x\in D_{+}(x_i)}\mathrm{Spec}({\mathcal{O}}_{X,x})$. For a point lying in two charts, [L1] identifies the point and the two local rings, so the decompositions agree on the overlap; since the standard charts cover $X$, this glues them into the finite disjoint union of the spectra of the local rings at all points of $X$.

Claim 1 is 1.1 and 3.1, claim 2 is 4.1, claim 3 is 4.2, and claim 4 is 4.3. The zero-dimensional hypothesis was used only through the chart rings $A_i$; the Axiom of Choice enters in the lying-over and chain-lifting suppliers of [L6], in the maximal-ideal and Artinian-structure suppliers of [L8], and it is the standing assumption [L9]. No finiteness of $X$ or Noetherianity was assumed in advance.