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  "body": "\n## Statement\n\nLet $A \\subseteq B \\subseteq C$ be commutative rings with $A$ a [[Noetherian ring|0670T59HGE]].\nSuppose that $C$ is a [[finitely generated|AC5TFFQT9M]] $A$-algebra and that $C$ is a [[finitely generated|05QY1VEBYD]] $B$-module.\nThen $B$ is a finitely generated $A$-algebra.\n\n## Remarks\n\nLet $x_1, \\ldots, x_m$ generate $C$ as an $A$-algebra and $y_1, \\ldots, y_n$ generate $C$ as a $B$-module.\nWrite $x_i = \\sum_j b_{ij} y_j$ and $y_i y_j = \\sum_l b_{ijl} y_l$ with $b_{ij}, b_{ijl} \\in B$.\nLet $B_0$ 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.\nThen $B$ 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 $A$.\n\nExample: 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.\n\n## Explanations\n\n### L1 human\n\nIf 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.\nFinite generation passes down to an intermediate ring as long as the top is finite over it.\n\n### L2 human\n\nThe lemma gives a short proof of [[Zariski's lemma|YBJ2VQHFDQ]].\nSuppose 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 $r \\ge 1$.\nThen $K$ 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.\nThe same idea shows that rings of invariants of finite groups acting on finitely generated algebras over a field are finitely generated.\n",
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