The Stacks project

31.7 Relative assassin

Let $A \to B$ be a ring map. Let $N$ be a $B$-module. Recall that a prime $\mathfrak q \subset B$ is said to be in the relative assassin of $N$ over $B/A$ if $\mathfrak q$ is an associated prime of $N \otimes _ A \kappa (\mathfrak p)$. Here $\mathfrak p = A \cap \mathfrak q$. See Algebra, Definition 10.65.2. Here is the definition of the relative assassin for quasi-coherent sheaves over a morphism of schemes.

Definition 31.7.1. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. The relative assassin of $\mathcal{F}$ in $X$ over $S$ is the set

\[ \text{Ass}_{X/S}(\mathcal{F}) = \bigcup \nolimits _{s \in S} \text{Ass}_{X_ s}(\mathcal{F}_ s) \]

where $\mathcal{F}_ s = (X_ s \to X)^*\mathcal{F}$ is the restriction of $\mathcal{F}$ to the fibre of $f$ at $s$.

Again there is a caveat that this is best used when the fibres of $f$ are locally Noetherian and $\mathcal{F}$ is of finite type. In the general case we should probably use the relative weak assassin (defined in the next section). Let us link the scheme theoretic notion with the algebraic notion on affine opens; note that this correspondence works perfectly only for morphisms of schemes whose fibres are locally Noetherian.

Lemma 31.7.2. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $U \subset X$ and $V \subset S$ be affine opens with $f(U) \subset V$. Write $U = \mathop{\mathrm{Spec}}(A)$, $V = \mathop{\mathrm{Spec}}(R)$, and set $M = \Gamma (U, \mathcal{F})$. Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime. Then

\[ \mathfrak p \in \text{Ass}_{A/R}(M) \Rightarrow x \in \text{Ass}_{X/S}(\mathcal{F}) \]

If all fibres $X_ s$ of $f$ are locally Noetherian, then $\mathfrak p \in \text{Ass}_{A/R}(M) \Leftrightarrow x \in \text{Ass}_{X/S}(\mathcal{F})$ for all pairs $(\mathfrak p, x)$ as above.

Proof. The set $\text{Ass}_{A/R}(M)$ is defined in Algebra, Definition 10.65.2. Choose a pair $(\mathfrak p, x)$. Let $s = f(x)$. Let $\mathfrak r \subset R$ be the prime lying under $\mathfrak p$, i.e., the prime corresponding to $s$. Let $\mathfrak p' \subset A \otimes _ R \kappa (\mathfrak r)$ be the prime whose inverse image is $\mathfrak p$, i.e., the prime corresponding to $x$ viewed as a point of its fibre $X_ s$. Then $\mathfrak p \in \text{Ass}_{A/R}(M)$ if and only if $\mathfrak p'$ is an associated prime of $M \otimes _ R \kappa (\mathfrak r)$, see Algebra, Lemma 10.65.1. Note that the ring $A \otimes _ R \kappa (\mathfrak r)$ corresponds to $U_ s$ and the module $M \otimes _ R \kappa (\mathfrak r)$ corresponds to the quasi-coherent sheaf $\mathcal{F}_ s|_{U_ s}$. Hence $x$ is an associated point of $\mathcal{F}_ s$ by Lemma 31.2.2. The reverse implication holds if $\mathfrak p'$ is finitely generated which is how the last sentence is seen to be true. $\square$

Lemma 31.7.3. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $g : S' \to S$ be a morphism of schemes. Consider the base change diagram

\[ \xymatrix{ X' \ar[d] \ar[r]_{g'} & X \ar[d] \\ S' \ar[r]^ g & S } \]

and set $\mathcal{F}' = (g')^*\mathcal{F}$. Let $x' \in X'$ be a point with images $x \in X$, $s' \in S'$ and $s \in S$. Assume $f$ locally of finite type. Then $x' \in \text{Ass}_{X'/S'}(\mathcal{F}')$ if and only if $x \in \text{Ass}_{X/S}(\mathcal{F})$ and $x'$ corresponds to a generic point of an irreducible component of $\mathop{\mathrm{Spec}}(\kappa (s') \otimes _{\kappa (s)} \kappa (x))$.

Proof. Consider the morphism $X'_{s'} \to X_ s$ of fibres. As $X_{s'} = X_ s \times _{\mathop{\mathrm{Spec}}(\kappa (s))} \mathop{\mathrm{Spec}}(\kappa (s'))$ this is a flat morphism. Moreover $\mathcal{F}'_{s'}$ is the pullback of $\mathcal{F}_ s$ via this morphism. As $X_ s$ is locally of finite type over the Noetherian scheme $\mathop{\mathrm{Spec}}(\kappa (s))$ we have that $X_ s$ is locally Noetherian, see Morphisms, Lemma 29.15.6. Thus we may apply Lemma 31.3.1 and we see that

\[ \text{Ass}_{X'_{s'}}(\mathcal{F}'_{s'}) = \bigcup \nolimits _{x \in \text{Ass}(\mathcal{F}_ s)} \text{Ass}((X'_{s'})_ x). \]

Thus to prove the lemma it suffices to show that the associated points of the fibre $(X'_{s'})_ x$ of the morphism $X'_{s'} \to X_ s$ over $x$ are its generic points. Note that $(X'_{s'})_ x = \mathop{\mathrm{Spec}}(\kappa (s') \otimes _{\kappa (s)} \kappa (x))$ as schemes. By Algebra, Lemma 10.167.1 the ring $\kappa (s') \otimes _{\kappa (s)} \kappa (x)$ is a Noetherian Cohen-Macaulay ring. Hence its associated primes are its minimal primes, see Algebra, Proposition 10.63.6 (minimal primes are associated) and Algebra, Lemma 10.157.2 (no embedded primes). $\square$

Remark 31.7.4. With notation and assumptions as in Lemma 31.7.3 we see that it is always the case that $(g')^{-1}(\text{Ass}_{X/S}(\mathcal{F})) \supset \text{Ass}_{X'/S'}(\mathcal{F}')$. If the morphism $S' \to S$ is locally quasi-finite, then we actually have

\[ (g')^{-1}(\text{Ass}_{X/S}(\mathcal{F})) = \text{Ass}_{X'/S'}(\mathcal{F}') \]

because in this case the field extensions $\kappa (s')/\kappa (s)$ are always finite. In fact, this holds more generally for any morphism $g : S' \to S$ such that all the field extensions $\kappa (s')/\kappa (s)$ are algebraic, because in this case all prime ideals of $\kappa (s') \otimes _{\kappa (s)} \kappa (x)$ are maximal (and minimal) primes, see Algebra, Lemma 10.36.19.

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