Lemma 71.3.1. 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 71.3.2. 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 71.3.3. 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 71.3.4. 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
$y \in \text{WeakAss}_ S(\mathcal{G})$,
$f$ is flat at $x$, and
the dimension of the local ring of the fibre of $f$ at $x$ is zero (Morphisms of Spaces, Definition 67.33.1),
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 66.4.3. 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 71.3.5. 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 71.3.6. 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 71.3.7. 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. $\square$
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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)