The Stacks project

Lemma 37.24.1. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\xi \in \text{Ass}_{X/S}(\mathcal{F})$ and set $Z = \overline{\{ \xi \} } \subset X$. If $f$ is locally of finite type and $\mathcal{F}$ is a finite type $\mathcal{O}_ X$-module, then there exists a nonempty open $V \subset Z$ such that for every $s \in f(V)$ the generic points of $V_ s$ are elements of $\text{Ass}_{X/S}(\mathcal{F})$.

Proof. We may replace $S$ by an affine open neighbourhood of $f(\xi )$ and $X$ by an affine open neighbourhood of $\xi $. Hence we may assume $S = \mathop{\mathrm{Spec}}(A)$, $X = \mathop{\mathrm{Spec}}(B)$ and that $f$ is given by the finite type ring map $A \to B$, see Morphisms, Lemma 29.15.2. Moreover, we may write $\mathcal{F} = \widetilde{M}$ for some finite $B$-module $M$, see Properties, Lemma 28.16.1. Let $\mathfrak q \subset B$ be the prime corresponding to $\xi $ and let $\mathfrak p \subset A$ be the corresponding prime of $A$. By assumption $\mathfrak q \in \text{Ass}_ B(M \otimes _ A \kappa (\mathfrak p))$, see Algebra, Remark 10.65.6 and Divisors, Lemma 31.2.2. With this notation $Z = V(\mathfrak q) \subset \mathop{\mathrm{Spec}}(B)$. In particular $f(Z) \subset V(\mathfrak p)$. Hence clearly it suffices to prove the lemma after replacing $A$, $B$, and $M$ by $A/\mathfrak pA$, $B/\mathfrak pB$, and $M/\mathfrak pM$. In other words we may assume that $A$ is a domain with fraction field $K$ and $\mathfrak q \subset B$ is an associated prime of $M \otimes _ A K$.

At this point we can use generic flatness. Namely, by Algebra, Lemma 10.118.3 there exists a nonzero $g \in A$ such that $M_ g$ is flat as an $A_ g$-module. After replacing $A$ by $A_ g$ we may assume that $M$ is flat as an $A$-module.

In this case, by Algebra, Lemma 10.65.4 we see that $\mathfrak q$ is also an associated prime of $M$. Hence we obtain an injective $B$-module map $B/\mathfrak q \to M$. Let $Q$ be the cokernel so that we obtain a short exact sequence

\[ 0 \to B/\mathfrak q \to M \to Q \to 0 \]

of finite $B$-modules. After applying generic flatness Algebra, Lemma 10.118.3 once more, this time to the $B$-module $Q$, we may assume that $Q$ is a flat $A$-module. In particular we may assume the short exact sequence above is universally injective, see Algebra, Lemma 10.39.12. In this situation $(B/\mathfrak q) \otimes _ A \kappa (\mathfrak p') \subset M \otimes _ A \kappa (\mathfrak p')$ for any prime $\mathfrak p'$ of $A$. The lemma follows as a minimal prime $\mathfrak q'$ of the support of $(B/\mathfrak q) \otimes _ A \kappa (\mathfrak p')$ is an associated prime of $(B/\mathfrak q) \otimes _ A \kappa (\mathfrak p')$ by Divisors, Lemma 31.2.9. $\square$


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 05F1. Beware of the difference between the letter 'O' and the digit '0'.