The Stacks project

70.3 Morphisms and weakly associated points

Lemma 70.3.1. Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then we have

\[ \text{WeakAss}_ S(f_*\mathcal{F}) \subset f(\text{WeakAss}_ X(\mathcal{F})) \]

Proof. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Set $U = X \times _ Y V$. Then $U \to V$ is an affine morphism of schemes. By our definition of weakly associated points the problem is reduced to the morphism of schemes $U \to V$. This case is treated in Divisors, Lemma 31.6.1. $\square$

Lemma 70.3.2. Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. If $X$ is locally Noetherian, then we have

\[ \text{WeakAss}_ Y(f_*\mathcal{F}) = f(\text{WeakAss}_ X(\mathcal{F})) \]

Proof. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Set $U = X \times _ Y V$. Then $U \to V$ is an affine morphism of schemes and $U$ is locally Noetherian. By our definition of weakly associated points the problem is reduced to the morphism of schemes $U \to V$. This case is treated in Divisors, Lemma 31.6.2. $\square$

Lemma 70.3.3. Let $S$ be a scheme. Let $f : X \to Y$ be a finite morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then $\text{WeakAss}(f_*\mathcal{F}) = f(\text{WeakAss}(\mathcal{F}))$.

Proof. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Set $U = X \times _ Y V$. Then $U \to V$ is a finite morphism of schemes. By our definition of weakly associated points the problem is reduced to the morphism of schemes $U \to V$. This case is treated in Divisors, Lemma 31.6.3. $\square$

Lemma 70.3.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ Y$-module. Let $x \in |X|$ and $y = f(x) \in |Y|$. If

  1. $y \in \text{WeakAss}_ S(\mathcal{G})$,

  2. $f$ is flat at $x$, and

  3. the dimension of the local ring of the fibre of $f$ at $x$ is zero (Morphisms of Spaces, Definition 66.33.1),

then $x \in \text{WeakAss}(f^*\mathcal{G})$.

Proof. Choose a scheme $V$, a point $v \in V$, and an étale morphism $V \to Y$ mapping $v$ to $y$. Choose a scheme $U$, a point $u \in U$, and an étale morphism $U \to V \times _ Y X$ mapping $v$ to a point lying over $v$ and $x$. This is possible because there is a $t \in |V \times _ Y X|$ mapping to $(v, y)$ by Properties of Spaces, Lemma 65.4.3. By definition we see that the dimension of $\mathcal{O}_{U_ v, u}$ is zero. Hence $u$ is a generic point of the fiber $U_ v$. By our definition of weakly associated points the problem is reduced to the morphism of schemes $U \to V$. This case is treated in Divisors, Lemma 31.6.4. $\square$

Lemma 70.3.5. Let $K/k$ be a field extension. Let $X$ be an algebraic space over $k$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $y \in X_ K$ with image $x \in X$. If $y$ is a weakly associated point of the pullback $\mathcal{F}_ K$, then $x$ is a weakly associated point of $\mathcal{F}$.

Proof. This is the translation of Divisors, Lemma 31.6.5 into the language of algebraic spaces. We omit the details of the translation. $\square$

Lemma 70.3.6. Let $S$ be a scheme. Let $f : X \to Y$ be a finite flat morphism of algebraic spaces. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ Y$-module. Let $x \in |X|$ be a point with image $y \in |Y|$. Then

\[ x \in \text{WeakAss}(g^*\mathcal{G}) \Leftrightarrow y \in \text{WeakAss}(\mathcal{G}) \]

Proof. Follows immediately from the case of schemes (More on Flatness, Lemma 38.2.7) by étale localization. $\square$

Lemma 70.3.7. Let $S$ be a scheme. Let $f : X \to Y$ be an étale morphism of algebraic spaces. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ Y$-module. Let $x \in |X|$ be a point with image $y \in |Y|$. Then

\[ x \in \text{WeakAss}(f^*\mathcal{G}) \Leftrightarrow y \in \text{WeakAss}(\mathcal{G}) \]

Proof. This is immediate from the definition of weakly associated points and in fact the corresponding lemma for the case of schemes (More on Flatness, Lemma 38.2.8) is the basis for our definition. $\square$


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