Section 31.3 (05DA): Morphisms and associated points

31.3 Morphisms and associated points

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. If $s \in S$ is a point, then it is often convenient to denote $\mathcal{F}_ s$ the $\mathcal{O}_{X_ s}$-module one gets by pulling back $\mathcal{F}$ by the morphism $i_ s : X_ s \to X$. Here $X_ s$ is the scheme theoretic fibre of $f$ over $s$. In a formula

\[ \mathcal{F}_ s = i_ s^*\mathcal{F} \]

Of course, this notation clashes with the already existing notation for the stalk of $\mathcal{F}$ at a point $x \in X$ if $f = \text{id}_ X$. However, the notation is often convenient, as in the formulation of the following lemma.

Lemma 31.3.1. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$ which is flat over $S$. Let $\mathcal{G}$ be a quasi-coherent sheaf on $S$. Then we have

\[ \text{Ass}_ X(\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G}) \supset \bigcup \nolimits _{s \in \text{Ass}_ S(\mathcal{G})} \text{Ass}_{X_ s}(\mathcal{F}_ s) \]

and equality holds if $S$ is locally Noetherian (for the notation $\mathcal{F}_ s$ see above).

Proof. Let $x \in X$ and let $s = f(x) \in S$. Set $B = \mathcal{O}_{X, x}$, $A = \mathcal{O}_{S, s}$, $N = \mathcal{F}_ x$, and $M = \mathcal{G}_ s$. Note that the stalk of $\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G}$ at $x$ is equal to the $B$-module $M \otimes _ A N$. Hence $x \in \text{Ass}_ X(\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G})$ if and only if $\mathfrak m_ B$ is in $\text{Ass}_ B(M \otimes _ A N)$. Similarly $s \in \text{Ass}_ S(\mathcal{G})$ and $x \in \text{Ass}_{X_ s}(\mathcal{F}_ s)$ if and only if $\mathfrak m_ A \in \text{Ass}_ A(M)$ and $\mathfrak m_ B/\mathfrak m_ A B \in \text{Ass}_{B \otimes \kappa (\mathfrak m_ A)}(N \otimes \kappa (\mathfrak m_ A))$. Thus the lemma follows from Algebra, Lemma 10.65.5. $\square$

Comments (2)

Comment #2782 by Claus on August 23, 2017 at 07:54

I find the notation $\mathcal{F}_s$ unfortunate for the following reason:

If $x \in X$ then $\mathcal{F}_x$ is the stalk at $x$ . Now take $f$ to be the identity, and then you have a clash of notations. Also more generally, I think that a notation like $\mathcal{F}_{x}, \mathcal{F}_y, \ldots$ should be reserved for stalks of sheaves.

I suggest to write $\mathcal{F}_{|X_s}$ what is $\mathcal{F}_s$ now.

Comment #2891 by Johan on October 07, 2017 at 14:55

Although you are right, I've chosen not to change the notation for now. I looked around and in most places where $\mathcal{F}_s$ is used, we say explicitly what we mean. Also, as long as it is clear that $s$ is a point of a different space than $X$ , confusion is unlikely to occur. I've fixed the problem with this particular lemma by explaining the notation clearly. Thanks! See changes here.

The Stacks project

31.3 Morphisms and associated points

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