The Stacks project

Lemma 87.14.4. Let $\varphi : A \to B$ be an arrow of $\textit{WAdm}^{Noeth}$ which is adic and topologically of finite type. Let $\mathfrak q \subset B$ be rig-closed. Let $\mathfrak p = \varphi ^{-1}(\mathfrak q) \subset A$. Let $\mathfrak a \subset A$ be the ideal of topologically nilpotent elements. The following are equivalent

  1. the residue field $\kappa $ of $B/\mathfrak q$ is finite over $A/\mathfrak a$,

  2. $\mathfrak p \subset A$ is rig-closed,

  3. $A/\mathfrak p \subset B/\mathfrak q$ is a finite extension of rings.

Proof. Assume (1). Recall that $B/\mathfrak q$ is a Noetherian local ring of dimension $1$ whose topology is the adic topology coming from the maximal ideal. Since $\varphi $ is adic, we see that $A \to B/\mathfrak q$ is adic. Hence $\varphi (\mathfrak a)$ is a nonzero ideal in $B/\mathfrak q$. Hence $B/\mathfrak q + \varphi (\mathfrak a)$ has finite length. Hence $B/\mathfrak q + \varphi (\mathfrak a)$ is finite as an $A/\mathfrak a$-module by our assumption. Thus $B/\mathfrak q$ is finite over $A$ by Algebra, Lemma 10.96.12. Thus (3) holds.

Assume (3). Then $\mathop{\mathrm{Spec}}(B/\mathfrak q) \to \mathop{\mathrm{Spec}}(A/\mathfrak p)$ is surjective by Algebra, Lemma 10.36.17. This implies (2).

Assume (2). Denote $\kappa '$ the residue field of $A/\mathfrak p$. By Lemma 87.14.3 (and Lemma 87.12.4) the extension $\kappa /\kappa '$ is finitely generated as an algebra. By the Hilbert Nullstellensatz (Algebra, Lemma 10.34.2) we see that $\kappa /\kappa '$ is a finite extension. Hence we see that (1) holds. $\square$

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