Lemma 23.10.2. Let $A \to B$ be a flat finite type ring map of Noetherian rings. If

is a perfect ring map, i.e., if $B$ has finite tor dimension over $B \otimes _ A B$, then $B$ is a smooth $A$-algebra.

Lemma 23.10.2. Let $A \to B$ be a flat finite type ring map of Noetherian rings. If

\[ B \otimes _ A B \longrightarrow B \]

is a perfect ring map, i.e., if $B$ has finite tor dimension over $B \otimes _ A B$, then $B$ is a smooth $A$-algebra.

**Proof.**
This follows from Lemma 23.10.1 and general facts about smooth ring maps, see Algebra, Lemmas 10.137.12 and 10.137.13. Alternatively, the reader can slightly modify the proof of Lemma 23.10.1 to prove this lemma.
$\square$

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