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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09QF. Beware of the difference between the letter 'O' and the digit '0'.