## 71.4 Relative weak assassin

We need a couple of lemmas to define this gadget.

Lemma 71.4.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $y \in |Y|$. The following are equivalent

1. for some scheme $V$, point $v \in V$, and étale morphism $V \to Y$ mapping $v$ to $y$, the algebraic space $X_ v$ is locally Noetherian,

2. for every scheme $V$, point $v \in V$, and étale morphism $V \to Y$ mapping $v$ to $y$, the algebraic space $X_ v$ is locally Noetherian, and

3. there exists a field $k$ and a morphism $\mathop{\mathrm{Spec}}(k) \to Y$ representing $y$ such that $X_ k$ is locally Noetherian.

If there exists a field $k_0$ and a monomorphism $\mathop{\mathrm{Spec}}(k_0) \to Y$ representing $y$, then these are also equivalent to

1. the algebraic space $X_{k_0}$ is locally Noetherian.

Proof. Observe that $X_ v = v \times _ Y X = \mathop{\mathrm{Spec}}(\kappa (v)) \times _ Y X$. Hence the implications (2) $\Rightarrow$ (1) $\Rightarrow$ (3) are clear. Assume that $\mathop{\mathrm{Spec}}(k) \to Y$ is a morphism from the spectrum of a field such that $X_ k$ is locally Noetherian. Let $V \to Y$ be an étale morphism from a scheme $V$ and let $v \in V$ a point mapping to $y$. Then the scheme $v \times _ Y \mathop{\mathrm{Spec}}(k)$ is nonempty. Choose a point $w \in v \times _ Y \mathop{\mathrm{Spec}}(k)$. Consider the morphisms

$X_ v \longleftarrow X_ w \longrightarrow X_ k$

Since $V \to Y$ is étale and since $w$ may be viewed as a point of $V \times _ Y \mathop{\mathrm{Spec}}(k)$, we see that $\kappa (w)/k$ is a finite separable extension of fields (Morphisms, Lemma 29.36.7). Thus $X_ w \to X_ k$ is a finite étale morphism as a base change of $w \to \mathop{\mathrm{Spec}}(k)$. Hence $X_ w$ is locally Noetherian (Morphisms of Spaces, Lemma 67.23.5). The morphism $X_ w \to X_ v$ is a surjective, affine, flat morphism as a base change of the surjective, affine, flat morphism $w \to v$. Then the fact that $X_ w$ is locally Noetherian implies that $X_ v$ is locally Noetherian. This can be seen by picking a surjective étale morphism $U \to X$ and then using that $U_ w \to U_ v$ is surjective, affine, and flat. Working affine locally on the scheme $U_ v$ we conclude that $U_ w$ is locally Noetherian by Algebra, Lemma 10.164.1.

Finally, it suffices to prove that (3) implies (4) in case we have a monomorphism $\mathop{\mathrm{Spec}}(k_0) \to Y$ in the class of $y$. Then $\mathop{\mathrm{Spec}}(k) \to Y$ factors as $\mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(k_0) \to Y$. The argument given above then shows that $X_ k$ being locally Noetherian impies that $X_{k_0}$ is locally Noetherian. $\square$

Definition 71.4.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $y \in |Y|$. We say the fibre of $f$ over $y$ is locally Noetherian if the equivalent conditions (1), (2), and (3) of Lemma 71.4.1 are satisfied. We say the fibres of $f$ are locally Noetherian if this holds for every $y \in |Y|$.

Of course, the usual way to guarantee locally Noetherian fibres is to assume the morphism is locally of finite type.

Lemma 71.4.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type, then the fibres of $f$ are locally Noetherian.

Proof. This follows from Morphisms of Spaces, Lemma 67.23.5 and the fact that the spectrum of a field is Noetherian. $\square$

Lemma 71.4.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $x \in |X|$ and $y = f(x) \in |Y|$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Consider commutative diagrams

$\xymatrix{ X \ar[d] & X \times _ Y V \ar[d] \ar[l] & X_ v \ar[d] \ar[l] \\ Y & V \ar[l] & v \ar[l] } \quad \xymatrix{ X \ar[d] & U \ar[d] \ar[l] & U_ v \ar[d] \ar[l] \\ Y & V \ar[l] & v \ar[l] } \quad \xymatrix{ x \ar@{|->}[d] & x' \ar@{|->}[d] \ar@{|->}[l] & u \ar@{|->}[ld] \ar@{|->}[l] \\ y & v \ar@{|->}[l] }$

where $V$ and $U$ are schemes, $V \to Y$ and $U \to X \times _ Y V$ are étale, $v \in V$, $x' \in |X_ v|$, $u \in U$ are points related as in the last diagram. Denote $\mathcal{F}|_{X_ v}$ and $\mathcal{F}|_{U_ v}$ the pullbacks of $\mathcal{F}$. The following are equivalent

1. for some $V, v, x'$ as above $x'$ is a weakly associated point of $\mathcal{F}|_{X_ v}$,

2. for every $V \to Y, v, x'$ as above $x'$ is a weakly associated point of $\mathcal{F}|_{X_ v}$,

3. for some $U, V, u, v$ as above $u$ is a weakly associated point of $\mathcal{F}|_{U_ v}$,

4. for every $U, V, u, v$ as above $u$ is a weakly associated point of $\mathcal{F}|_{U_ v}$,

5. for some field $k$ and morphism $\mathop{\mathrm{Spec}}(k) \to Y$ representing $y$ and some $t \in |X_ k|$ mapping to $x$, the point $t$ is a weakly associated point of $\mathcal{F}|_{X_ k}$.

If there exists a field $k_0$ and a monomorphism $\mathop{\mathrm{Spec}}(k_0) \to Y$ representing $y$, then these are also equivalent to

1. $x_0$ is a weakly associated point of $\mathcal{F}|_{X_{k_0}}$ where $x_0 \in |X_{k_0}|$ is the unique point mapping to $x$.

If the fibre of $f$ over $y$ is locally Noetherian, then in conditions (1), (2), (3), (4), and (6) we may replace “weakly associated” with “associated”.

Proof. Observe that given $V, v, x'$ as in the lemma we can find $U \to X \times _ Y V$ and $u \in U$ mapping to $x'$ and then the morphism $U_ v \to X_ v$ is étale. Thus it is clear that (1) and (3) are equivalent as well as (2) and (4). Each of these implies (5). We will show that (5) implies (2). Suppose given $V, v, x'$ as well as $\mathop{\mathrm{Spec}}(k) \to X$ and $t \in |X_ k|$ such that the point $t$ is a weakly associated point of $\mathcal{F}|_{X_ k}$. We can choose a point $w \in v \times _ Y \mathop{\mathrm{Spec}}(k)$. Then we obtain the morphisms

$X_ v \longleftarrow X_ w \longrightarrow X_ k$

Since $V \to Y$ is étale and since $w$ may be viewed as a point of $V \times _ Y \mathop{\mathrm{Spec}}(k)$, we see that $\kappa (w)/k$ is a finite separable extension of fields (Morphisms, Lemma 29.36.7). Thus $X_ w \to X_ k$ is a finite étale morphism as a base change of $w \to \mathop{\mathrm{Spec}}(k)$. Thus any point $x''$ of $X_ w$ lying over $t$ is a weakly associated point of $\mathcal{F}|_{X_ w}$ by Lemma 71.3.7. We may pick $x''$ mapping to $x'$ (Properties of Spaces, Lemma 66.4.3). Then Lemma 71.3.5 implies that $x'$ is a weakly associated point of $\mathcal{F}|_{X_ v}$.

To finish the proof it suffices to show that the equivalent conditions (1) – (5) imply (6) if we are given $\mathop{\mathrm{Spec}}(k_0) \to Y$ as in (6). In this case the morphism $\mathop{\mathrm{Spec}}(k) \to Y$ of (5) factors uniquely as $\mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(k_0) \to Y$. Then $x_0$ is the image of $t$ under the morphism $X_ k \to X_{k_0}$. Hence the same lemma as above shows that (6) is true. $\square$

Definition 71.4.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. The relative weak assassin of $\mathcal{F}$ in $X$ over $Y$ is the set $\text{WeakAss}_{X/Y}(\mathcal{F}) \subset |X|$ consisting of those $x \in |X|$ such that the equivalent conditions of Lemma 71.4.4 are satisfied. If the fibres of $f$ are locally Noetherian (Definition 71.4.2) then we use the notation $\text{Ass}_{X/Y}(\mathcal{F})$.

With this notation we can formulate some of the results already proven for schemes.

Lemma 71.4.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ Y$-module. Assume

1. $\mathcal{F}$ is flat over $Y$,

2. $X$ and $Y$ are locally Noetherian, and

3. the fibres of $f$ are locally Noetherian.

Then

$\text{Ass}_ X(\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G}) = \{ x \in \text{Ass}_{X/Y}(\mathcal{F})\text{ such that } f(x) \in \text{Ass}_ Y(\mathcal{G}) \}$

Proof. Via étale localization, this is an immediate consequence of the result for schemes, see Divisors, Lemma 31.3.1. The result for schemes is more general only because we haven't defined associated points for non-Noetherian algebraic spaces (hence we need to assume $X$ and the fibres of $X \to Y$ are locally Noetherian to even be able to formulate this result). $\square$

Lemma 71.4.7. Let $S$ be a scheme. Let

$\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y }$

be a cartesian diagram of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module and set $\mathcal{F}' = (g')^*\mathcal{F}$. If $f$ is locally of finite type, then

1. $x' \in \text{Ass}_{X'/Y'}(\mathcal{F}') \Rightarrow g'(x') \in \text{Ass}_{X/Y}(\mathcal{F})$

2. if $x \in \text{Ass}_{X/Y}(\mathcal{F})$, then given $y' \in |Y'|$ with $f(x) = g(y')$, there exists an $x' \in \text{Ass}_{X'/Y'}(\mathcal{F}')$ with $g'(x') = x$ and $f'(x') = y'$.

Proof. This follows from the case of schemes by étale localization. We write out the details completely. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. Choose a scheme $V'$ and a surjective étale morphism $V' \to V \times _ Y Y'$. Then $U' = V' \times _ V U$ is a scheme and the morphism $U' \to X'$ is surjective and étale.

Proof of (1). Choose $u' \in U'$ mapping to $x'$. Denote $v' \in V'$ the image of $u'$. Then $x' \in \text{Ass}_{X'/Y'}(\mathcal{F}')$ is equivalent to $u' \in \text{Ass}(\mathcal{F}|_{U'_{v'}})$ by definition (writing $\text{Ass}$ instead of $\text{WeakAss}$ makes sense as $U'_{v'}$ is locally Noetherian). Applying Divisors, Lemma 31.7.3 we see that the image $u \in U$ of $u'$ is in $\text{Ass}(\mathcal{F}|_{U_ v})$ where $v \in V$ is the image of $u$. This in turn means $g'(x') \in \text{Ass}_{X/Y}(\mathcal{F})$.

Proof of (2). Choose $u \in U$ mapping to $x$. Denote $v \in V$ the image of $u$. Then $x \in \text{Ass}_{X/Y}(\mathcal{F})$ is equivalent to $u \in \text{Ass}(\mathcal{F}|_{U_ v})$ by definition. Choose a point $v' \in V'$ mapping to $y' \in |Y'|$ and to $v \in V$ (possible by Properties of Spaces, Lemma 66.4.3). Let $t \in \mathop{\mathrm{Spec}}(\kappa (v') \otimes _{\kappa (v)} \kappa (u))$ be a generic point of an irreducible component. Let $u' \in U'$ be the image of $t$. Applying Divisors, Lemma 31.7.3 we see that $u' \in \text{Ass}(\mathcal{F}'|_{U'_{v'}})$. This in turn means $x' \in \text{Ass}_{X'/Y'}(\mathcal{F}')$ where $x' \in |X'|$ is the image of $u'$. $\square$

Lemma 71.4.8. With notation and assumptions as in Lemma 71.4.7. Assume $g$ is locally quasi-finite, or more generally that for every $y' \in |Y'|$ the transcendence degree of $y'/g(y')$ is $0$. Then $\text{Ass}_{X'/Y'}(\mathcal{F}')$ is the inverse image of $\text{Ass}_{X/Y}(\mathcal{F})$.

Proof. The transcendence degree of a point over its image is defined in Morphisms of Spaces, Definition 67.33.1. Let $x' \in |X'|$ with image $x \in |X|$. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. Choose a scheme $V'$ and a surjective étale morphism $V' \to V \times _ Y Y'$. Then $U' = V' \times _ V U$ is a scheme and the morphism $U' \to X'$ is surjective and étale. Choose $u \in U$ mapping to $x$. Denote $v \in V$ the image of $u$. Then $x \in \text{Ass}_{X/Y}(\mathcal{F})$ is equivalent to $u \in \text{Ass}(\mathcal{F}|_{U_ v})$ by definition. Choose a point $u' \in U'$ mapping to $x' \in |X'|$ and to $u \in U$ (possible by Properties of Spaces, Lemma 66.4.3). Let $v' \in V'$ be the image of $u'$. Then $x' \in \text{Ass}_{X'/Y'}(\mathcal{F}')$ is equivalent to $u' \in \text{Ass}(\mathcal{F}'|_{U'_{v'}})$ by definition. Now the lemma follows from the discussion in Divisors, Remark 31.7.4 applied to $u' \in \mathop{\mathrm{Spec}}(\kappa (v') \otimes _{\kappa (v)} \kappa (u))$. $\square$

Lemma 71.4.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $i : Z \to X$ be a finite morphism. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ Z$-module. Then $\text{WeakAss}_{X/Y}(i_*\mathcal{G}) = i(\text{WeakAss}_{Z/Y}(\mathcal{G}))$.

Proof. Follows from the case of schemes (Divisors, Lemma 31.8.3) by étale localization. Details omitted. $\square$

Lemma 71.4.10. Let $Y$ be a scheme. Let $X$ be an algebraic space of finite presentation over $Y$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite presentation. Let $U \subset X$ be an open subspace such that $U \to Y$ is quasi-compact. Then the set

$E = \{ y \in Y \mid \text{Ass}_{X_ y}(\mathcal{F}_ y) \subset |U_ y|\}$

is locally constructible in $Y$.

Proof. Note that since $Y$ is a scheme, it makes sense to take the fibres $X_ y = \mathop{\mathrm{Spec}}(\kappa (y)) \times _ Y X$. (Also, by our definitions, the set $\text{Ass}_{X_ y}(\mathcal{F}_ y)$ is exactly the fibre of $\text{Ass}_{X/Y}(\mathcal{F}) \to Y$ over $y$, but we won't need this.) The question is local on $Y$, indeed, we have to show that $E$ is constructible if $Y$ is affine. In this case $X$ is quasi-compact. Choose an affine scheme $W$ and a surjective étale morphism $\varphi : W \to X$. Then $\text{Ass}_{X_ y}(\mathcal{F}_ y)$ is the image of $\text{Ass}_{W_ y}(\varphi ^*\mathcal{F}_ y)$ for all $y \in Y$. Hence the lemma follows from the case of schemes for the open $\varphi ^{-1}(U) \subset W$ and the morphism $W \to Y$. The case of schemes is More on Morphisms, Lemma 37.25.5. $\square$

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