---
tag: 4MBR0W0S6N
kind: statement
role: lemma
name: Artin–Tate lemma
msc: ["13E15", "13E05"]
uses: [0670T59HGE, AC5TFFQT9M, 05QY1VEBYD, 0YKA9J3XEE]
about: [AC5TFFQT9M]
sources:
  - url: https://en.wikipedia.org/wiki/Artin–Tate_lemma
same_as:
  wikidata: Q19595940
  wikipedia: Artin–Tate lemma
---

## Statement

Let $A \subseteq B \subseteq C$ be commutative rings with $A$ a [[Noetherian ring|0670T59HGE]].
Suppose that $C$ is a [[finitely generated|AC5TFFQT9M]] $A$-algebra and that $C$ is a [[finitely generated|05QY1VEBYD]] $B$-module.
Then $B$ is a finitely generated $A$-algebra.

## Remarks

Let $x_1, \ldots, x_m$ generate $C$ as an $A$-algebra and $y_1, \ldots, y_n$ generate $C$ as a $B$-module.
Write $x_i = \sum_j b_{ij} y_j$ and $y_i y_j = \sum_l b_{ijl} y_l$ with $b_{ij}, b_{ijl} \in B$.
Let $B_0$ be the $A$-algebra generated by all $b_{ij}$ and $b_{ijl}$; it is Noetherian by [[Hilbert's basis theorem|0YKA9J3XEE]], and $C$ is a finitely generated $B_0$-module.
Then $B$ is a $B_0$-submodule of $C$, hence finitely generated as a module over the Noetherian ring $B_0$, so $B$ is of finite type over $A$.

Example: let $A = k$ be a field, $C = k[x]$ and $B = k[x^2]$; then $C$ is generated by $1$ and $x$ as a $B$-module, and indeed $B$ is generated by $x^2$ as a $k$-algebra.

## Explanations

### L1 human

If a big ring is finitely generated over a small one, and is a finite module over a middle ring, then the middle ring is finitely generated too.
Finite generation passes down to an intermediate ring as long as the top is finite over it.

### L2 human

The lemma gives a short proof of [[Zariski's lemma|YBJ2VQHFDQ]].
Suppose a field $K$ is a finitely generated $k$-algebra but not finite over $k$, and choose a transcendence basis $t_1, \ldots, t_r$ with $r \ge 1$.
Then $K$ is finite over $k(t_1, \ldots, t_r)$, and the lemma would make $k(t_1, \ldots, t_r)$ a finitely generated $k$-algebra, which is impossible.
The same idea shows that rings of invariants of finite groups acting on finitely generated algebras over a field are finitely generated.