The Stacks project

Lemma 33.20.2. Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme of dimension $0$. Then $X$ is a disjoint union of spectra of local Artinian $k$-algebras $A$ with $\dim _ k(A) < \infty $. If $X$ is an algebraic $k$-scheme of dimension $0$, then in addition $X$ is affine and the morphism $X \to \mathop{\mathrm{Spec}}(k)$ is finite.

Proof. Let $X$ be a locally algebraic $k$-scheme of dimension $0$. Let $U = \mathop{\mathrm{Spec}}(A) \subset X$ be an affine open subscheme. Since $\dim (X) = 0$ we see that $\dim (A) = 0$. By Noether normalization, see Algebra, Lemma 10.115.4 we see that there exists a finite injection $k \to A$, i.e., $\dim _ k(A) < \infty $. Hence $A$ is Artinian, see Algebra, Lemma 10.53.2. This implies that $A = A_1 \times \ldots \times A_ r$ is a product of finitely many Artinian local rings, see Algebra, Lemma 10.53.6. Of course $\dim _ k(A_ i) < \infty $ for each $i$ as the sum of these dimensions equals $\dim _ k(A)$.

The arguments above show that $X$ has an open covering whose members are finite discrete topological spaces. Hence $X$ is a discrete topological space. It follows that $X$ is isomorphic to the disjoint union of its connected components each of which is a singleton. Since a singleton scheme is affine we conclude (by the results of the paragraph above) that each of these singletons is the spectrum of a local Artinian $k$-algebra $A$ with $\dim _ k(A) < \infty $.

Finally, if $X$ is an algebraic $k$-scheme of dimension $0$, then $X$ is quasi-compact hence is a finite disjoint union $X = \mathop{\mathrm{Spec}}(A_1) \amalg \ldots \amalg \mathop{\mathrm{Spec}}(A_ r)$ hence affine (see Schemes, Lemma 26.6.8) and we have seen the finiteness of $X \to \mathop{\mathrm{Spec}}(k)$ in the first paragraph of the proof. $\square$

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