The Stacks project

Proof. It is immediate from the definitions that (1) is equivalent to (4) by looking at part (5) of Lemma 10.26.5. Recall that $\mathop{\mathrm{Spec}}(S)$ is sober, Noetherian, and Jacobson, see Lemmas 10.26.2, 10.31.5, 10.35.2, and 10.35.4. If $S$ has dimension $0$, then every point defines an irreducible component and there are only a finite number of irreducible components (Topology, Lemma 5.9.2). Thus (1) implies (2). Trivially (2) implies (3). If (3) holds, then $\mathop{\mathrm{Spec}}(S)$ is discrete by Topology, Lemma 5.18.6 and hence the dimension of $S$ is $0$.

At this point we know that (1) – (4) are equivalent. The implication (5) $\Rightarrow $ (6) is Lemma 10.53.2. The implication (6) $\Rightarrow $ (7) follows from Proposition 10.60.7. The implication (7) $\Rightarrow $ (4) is immediate. Conversely, if $S$ satisfies (1) – (4), then $S$ has finitely many primes $\mathfrak m_1, \ldots , \mathfrak m_ r$ all maximal. Note that $\kappa (\mathfrak m_ i)$ is a finite extension of $k$ by the Hilbert Nullstellensatz (Theorem 10.34.1). By Proposition 10.60.7 we also see that $S$ is Artinian. Next, Lemma 10.53.6 tells us that $\text{length}_ S(S) < \infty $. Thus $\dim _ k(S) < \infty $ by Lemma 10.52.12. We conclude that (1) – (7) are equivalent. (Note: another and more standard way to prove $\dim (S) = 0 \Rightarrow \dim _ k(S) < \infty $ is to use Noether normalization, but we don't have this available to us yet.) $\square$