The Stacks project

Lemma 49.2.8. Let $A \to B$ be a quasi-finite homomorphism of Noetherian rings. Then $\text{Ass}_ B(\omega _{B/A})$ is the set of primes of $B$ lying over associated primes of $A$.

Proof. Choose a factorization $A \to B' \to B$ with $A \to B'$ finite and $B' \to B$ inducing an open immersion on spectra. As $\omega _{B/A} = \omega _{B'/A} \otimes _{B'} B$ it suffices to prove the statement for $\omega _{B'/A}$. Thus we may assume $A \to B$ is finite.

Assume $\mathfrak p \in \text{Ass}(A)$ and $\mathfrak q$ is a prime of $B$ lying over $\mathfrak p$. Let $x \in A$ be an element whose annihilator is $\mathfrak p$. Choose a nonzero $\kappa (\mathfrak p)$ linear map $\lambda : \kappa (\mathfrak q) \to \kappa (\mathfrak p)$. Since $A/\mathfrak p \subset B/\mathfrak q$ is a finite extension of rings, there is an $f \in A$, $f \not\in \mathfrak p$ such that $f\lambda $ maps $B/\mathfrak q$ into $A/\mathfrak p$. Hence we obtain a nonzero $A$-linear map

\[ B \to B/\mathfrak q \to A/\mathfrak p \to A,\quad b \mapsto f\lambda (b)x \]

An easy computation shows that this element of $\omega _{B/A}$ has annihilator $\mathfrak q$, whence $\mathfrak q \in \text{Ass}(\omega _{B/A})$.

Conversely, suppose that $\mathfrak q \subset B$ is a prime ideal lying over a prime $\mathfrak p \subset A$ which is not an associated prime of $A$. We have to show that $\mathfrak q \not\in \text{Ass}_ B(\omega _{B/A})$. After replacing $A$ by $A_\mathfrak p$ and $B$ by $B_\mathfrak p$ we may assume that $\mathfrak p$ is a maximal ideal of $A$. This is allowed by Lemma 49.2.5 and Algebra, Lemma 10.63.16. Then there exists an $f \in \mathfrak m$ which is a nonzerodivisor on $A$. Then $f$ is a nonzerodivisor on $\omega _{B/A}$ and hence $\mathfrak q$ is not an associated prime of this module. $\square$

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Your email address will not be published.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 0BVD. Beware of the difference between the letter 'O' and the digit '0'.