bibkey: artintate1951finiteringextensions
authors: Emil Artin; John T. Tate
year: 1951
title: "A Note on Finite Ring Extensions"
doi: 10.2969/jmsj/00310074
url: https://stacks.math.columbia.edu/tag/00IS
claim: "If R is a Noetherian ring, S a finitely generated R-algebra and T an R-subalgebra of S over which S is a finitely generated module, then T is a finitely generated R-algebra. Applied with R a field k and S = k[X], it gives that every k-subalgebra of k[X] is finitely generated."
strata_touched:
  - D5/S3/Topology/KnotFamilyVassilievFiniteGeneration
license: citation-only
triage: anchor

The Artin–Tate lemma

E. Artin and J. T. Tate, “A Note on Finite Ring Extensions”, Journal of the Mathematical Society of Japan 3(1) (1951), 74–77. The lemma is restated as Lemma 10.51.7 (tag 00IS) of the Stacks Project, Chapter 10 (Commutative Algebra), Section 10.51:

Let $R$ be a Noetherian ring. Let $S$ be a finitely generated $R$-algebra. If $T\subset S$ is an $R$-subalgebra such that $S$ is finitely generated as a $T$-module, then $T$ is of finite type over $R$.

Pinned Mathlib states it as Algebra.fg_of_fg_of_fg (Mathlib/RingTheory/Adjoin/Tower.lean), with the references Atiyah–Macdonald Proposition 7.8, Altman–Kleiman 16.17 and Stacks tag 00IS.

The consequence used

Let $k$ be a field and $T$ a $k$-subalgebra of $k[X]$. If $T$ consists of constants, it is generated by the empty set. Otherwise $T$ contains a nonconstant $f$ with leading coefficient $c\ne 0$, and $X$ is a root of the monic polynomial $c^{-1}(f(Y)-f)$ in $Y$ with coefficients in $k[f]\subseteq T$. So $k[X]$ is generated by $X$ as an algebra and integral over $T$, hence a finitely generated $T$-module. The lemma with $R=k$ and $S=k[X]$ shows that $T$ is a finitely generated $k$-algebra. This derivation is written out here; it is not a claim of verbatim appearance in the cited sources.

Verified locator

• DOI: 10.2969/jmsj/00310074 (https://doi.org/10.2969/jmsj/00310074; Crossref lists Journal of the Mathematical Society of Japan, volume 3, issue 1, pages 74–77, published 1 May 1951, authors Emil Artin and John T. Tate).
• URL: https://stacks.math.columbia.edu/tag/00IS (Lemma 10.51.7, “Artin-Tate”; the statement above is quoted from this page).
• The interior of the 1951 paper was not read here; the statement is attested by the Stacks Project page and the Mathlib docstring.