The Stacks project

Lemma 23.9.6. Let $A$ be a Noetherian ring. Let $A \to B$ be a finite type ring map such that the image of $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ contains all closed points of $\mathop{\mathrm{Spec}}(A)$. Then the following are equivalent

  1. $B$ is a complete intersection and $A \to B$ has finite tor dimension,

  2. $A$ is a complete intersection and $A \to B$ is a local complete intersection in the sense of More on Algebra, Definition 15.33.2.

Proof. This is a reformulation of Proposition 23.9.2 via Lemma 23.9.5. We omit the details. $\square$

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