The Stacks project

Lemma 15.82.4. A ring map which is flat and of finite presentation is perfect.

Proof. Let $A \to B$ be a ring map which is flat and of finite presentation. It is clear that $B$ has finite tor dimension. By Algebra, Lemma 10.168.1 there exists a finite type $\mathbf{Z}$-algebra $A_0 \subset A$ and a flat finite type ring map $A_0 \to B_0$ such that $B = B_0 \otimes _{A_0} A$. By Lemma 15.81.17 we see that $A_0 \to B_0$ is pseudo-coherent. As $A_0 \to B_0$ is flat we see that $B_0$ and $A$ are tor independent over $A_0$, hence we may use Lemma 15.81.12 to conclude that $A \to B$ is pseudo-coherent. $\square$

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