The Stacks project

38.18 A criterion for purity

We first prove that given a flat family of finite type quasi-coherent sheaves the points in the relative assassin specialize to points in the relative assassins of nearby fibres (if they specialize at all).

Lemma 38.18.1. Let $f : X \to S$ be a morphism of schemes of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. Let $s \in S$. Assume that $\mathcal{F}$ is flat over $S$ at all points of $X_ s$. Let $x' \in \text{Ass}_{X/S}(\mathcal{F})$ with $f(x') = s'$ such that $s' \leadsto s$ is a specialization in $S$. If $x'$ specializes to a point of $X_ s$, then $x' \leadsto x$ with $x \in \text{Ass}_{X_ s}(\mathcal{F}_ s)$.

Proof. Let $x' \leadsto t$ be a specialization with $t \in X_ s$. We may replace $X$ by an affine neighbourhood of $t$ and $S$ by an affine neighbourhood of $s$. Choose a closed immersion $i : X \to \mathbf{A}^ n_ S$. Then it suffices to prove the lemma for the module $i_*\mathcal{F}$ on $\mathbf{A}^ n_ S$ and the point $i(x')$. Hence we may assume $X \to S$ is of finite presentation.

Let $x' \leadsto t$ be a specialization with $t \in X_ s$. Set $A = \mathcal{O}_{S, s}$, $B = \mathcal{O}_{X, t}$, and $N = \mathcal{F}_ t$. Note that $B$ is essentially of finite presentation over $A$ and that $N$ is a finite $B$-module flat over $A$. Also $N$ is a finitely presented $B$-module by Lemma 38.10.9. Let $\mathfrak q' \subset B$ be the prime ideal corresponding to $x'$ and let $\mathfrak p' \subset A$ be the prime ideal corresponding to $s'$. The assumption $x' \in \text{Ass}_{X/S}(\mathcal{F})$ means that $\mathfrak q'$ is an associated prime of $N \otimes _ A \kappa (\mathfrak p')$. Let $\Sigma \subset B$ be the multiplicative subset of elements which are not zerodivisors on $N/\mathfrak m_ A N$. By Lemma 38.7.2 the map $N \to \Sigma ^{-1}N$ is universally injective. In particular, we see that $N \otimes _ A \kappa (\mathfrak p') \to \Sigma ^{-1}N \otimes _ A \kappa (\mathfrak p')$ is injective which implies that $\mathfrak q'$ is an associated prime of $\Sigma ^{-1}N \otimes _ A \kappa (\mathfrak p')$ and hence $\mathfrak q'$ is in the image of $\mathop{\mathrm{Spec}}(\Sigma ^{-1}B) \to \mathop{\mathrm{Spec}}(B)$. Thus Lemma 38.7.1 implies that $\mathfrak q' \subset \mathfrak q$ for some prime $\mathfrak q \in \text{Ass}_ B(N/\mathfrak m_ A N)$ (which in particular implies that $\mathfrak m_ A = A \cap \mathfrak q$). If $x \in X_ s$ denotes the point corresponding to $\mathfrak q$, then $x \in \text{Ass}_{X_ s}(\mathcal{F}_ s)$ and $x' \leadsto x$ as desired. $\square$

Lemma 38.18.2. Let $f : X \to S$ be a morphism of schemes of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. Let $s \in S$. Let $(S', s') \to (S, s)$ be an elementary étale neighbourhood and let

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

be a commutative diagram of morphisms of schemes. Assume

  1. $\mathcal{F}$ is flat over $S$ at all points of $X_ s$,

  2. $X' \to S'$ is of finite type,

  3. $g^*\mathcal{F}$ is pure along $X'_{s'}$,

  4. $g : X' \to X$ is étale, and

  5. $g(X')$ contains $\text{Ass}_{X_ s}(\mathcal{F}_ s)$.

In this situation $\mathcal{F}$ is pure along $X_ s$ if and only if the image of $X' \to X \times _ S S'$ contains the points of $\text{Ass}_{X \times _ S S'/S'}(\mathcal{F} \times _ S S')$ lying over points in $S'$ which specialize to $s'$.

Proof. Since the morphism $S' \to S$ is étale, we see that if $\mathcal{F}$ is pure along $X_ s$, then $\mathcal{F} \times _ S S'$ is pure along $X_ s$, see Lemma 38.16.4. Since purity satisfies flat descent, see Lemma 38.16.6, we see that if $\mathcal{F} \times _ S S'$ is pure along $X_{s'}$, then $\mathcal{F}$ is pure along $X_ s$. Hence we may replace $S$ by $S'$ and assume that $S = S'$ so that $g : X' \to X$ is an étale morphism between schemes of finite type over $S$. Moreover, we may replace $S$ by $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ and assume that $S$ is local.

First, assume that $\mathcal{F}$ is pure along $X_ s$. In this case every point of $\text{Ass}_{X/S}(\mathcal{F})$ specializes to a point of $X_ s$ by purity. Hence by Lemma 38.18.1 we see that every point of $\text{Ass}_{X/S}(\mathcal{F})$ specializes to a point of $\text{Ass}_{X_ s}(\mathcal{F}_ s)$. Thus every point of $\text{Ass}_{X/S}(\mathcal{F})$ is in the image of $g$ (as the image is open and contains $\text{Ass}_{X_ s}(\mathcal{F}_ s)$).

Conversely, assume that $g(X')$ contains $\text{Ass}_{X/S}(\mathcal{F})$. Let $S^ h = \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^ h)$ be the henselization of $S$ at $s$. Denote $g^ h : (X')^ h \to X^ h$ the base change of $g$ by $S^ h \to S$, and denote $\mathcal{F}^ h$ the pullback of $\mathcal{F}$ to $X^ h$. By Divisors, Lemma 31.7.3 and Remark 31.7.4 the relative assassin $\text{Ass}_{X^ h/S^ h}(\mathcal{F}^ h)$ is the inverse image of $\text{Ass}_{X/S}(\mathcal{F})$ via the projection $X^ h \to X$. As we have assumed that $g(X')$ contains $\text{Ass}_{X/S}(\mathcal{F})$ we conclude that the base change $g^ h((X')^ h) = g(X') \times _ S S^ h$ contains $\text{Ass}_{X^ h/S^ h}(\mathcal{F}^ h)$. In this way we reduce to the case where $S$ is the spectrum of a henselian local ring. Let $x \in \text{Ass}_{X/S}(\mathcal{F})$. To finish the proof of the lemma we have to show that $x$ specializes to a point of $X_ s$, see criterion (4) for purity in discussion following Definition 38.16.1. By assumption there exists a $x' \in X'$ such that $g(x') = x$. As $g : X' \to X$ is étale, we see that $x' \in \text{Ass}_{X'/S}(g^*\mathcal{F})$, see Lemma 38.2.8 (applied to the morphism of fibres $X'_ w \to X_ w$ where $w \in S$ is the image of $x'$). Since $g^*\mathcal{F}$ is pure along $X'_ s$ we see that $x' \leadsto y$ for some $y \in X'_ s$. Hence $x = g(x') \leadsto g(y)$ and $g(y) \in X_ s$ as desired. $\square$

Lemma 38.18.3. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $s \in S$. Assume

  1. $f$ is of finite type,

  2. $\mathcal{F}$ is of finite type,

  3. $\mathcal{F}$ is flat over $S$ at all points of $X_ s$, and

  4. $\mathcal{F}$ is pure along $X_ s$.

Then $\mathcal{F}$ is universally pure along $X_ s$.

Proof. We first make a preliminary remark. Suppose that $(S', s') \to (S, s)$ is an elementary étale neighbourhood. Denote $\mathcal{F}'$ the pullback of $\mathcal{F}$ to $X' = X \times _ S S'$. By the discussion following Definition 38.16.1 we see that $\mathcal{F}'$ is pure along $X'_{s'}$. Moreover, $\mathcal{F}'$ is flat over $S'$ along $X'_{s'}$. Then it suffices to prove that $\mathcal{F}'$ is universally pure along $X'_{s'}$. Namely, given any morphism $(T, t) \to (S, s)$ of pointed schemes the fibre product $(T', t') = (T \times _ S S', (t, s'))$ is flat over $(T, t)$ and hence if $\mathcal{F}_{T'}$ is pure along $X_{t'}$ then $\mathcal{F}_ T$ is pure along $X_ t$ by Lemma 38.16.6. Thus during the proof we may always replace $(s, S)$ by an elementary étale neighbourhood. We may also replace $S$ by $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ due to the local nature of the problem.

Choose an elementary étale neighbourhood $(S', s') \to (S, s)$ and a commutative diagram

\[ \xymatrix{ X \ar[d] & X' \ar[l]^ g \ar[d] \\ S & \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \ar[l] } \]

such that $X' \to X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ is étale, $X_ s = g((X')_{s'})$, the scheme $X'$ is affine, and such that $\Gamma (X', g^*\mathcal{F})$ is a free $\mathcal{O}_{S', s'}$-module, see Lemma 38.12.11. Note that $X' \to \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ is of finite type (as a quasi-compact morphism which is the composition of an étale morphism and the base change of a finite type morphism). By our preliminary remarks in the first paragraph of the proof we may replace $S$ by $\mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$. Hence we may assume there exists a commutative diagram

\[ \xymatrix{ X \ar[dr] & & X' \ar[ll]^ g \ar[ld] \\ & S & } \]

of schemes of finite type over $S$, where $g$ is étale, $X_ s \subset g(X')$, with $S$ local with closed point $s$, with $X'$ affine, and with $\Gamma (X', g^*\mathcal{F})$ a free $\Gamma (S, \mathcal{O}_ S)$-module. Note that in this case $g^*\mathcal{F}$ is universally pure over $S$, see Lemma 38.17.4.

In this situation we apply Lemma 38.18.2 to deduce that $\text{Ass}_{X/S}(\mathcal{F}) \subset g(X')$ from our assumption that $\mathcal{F}$ is pure along $X_ s$ and flat over $S$ along $X_ s$. By Divisors, Lemma 31.7.3 and Remark 31.7.4 we see that for any morphism of pointed schemes $(T, t) \to (S, s)$ we have

\[ \text{Ass}_{X_ T/T}(\mathcal{F}_ T) \subset (X_ T \to X)^{-1}(\text{Ass}_{X/S}(\mathcal{F})) \subset g(X') \times _ S T = g_ T(X'_ T). \]

Hence by Lemma 38.18.2 applied to the base change of our displayed diagram to $(T, t)$ we conclude that $\mathcal{F}_ T$ is pure along $X_ t$ as desired. $\square$

Lemma 38.18.4. Let $f : X \to S$ be a finite type morphism of schemes. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Assume $\mathcal{F}$ is flat over $S$. In this case $\mathcal{F}$ is pure relative to $S$ if and only if $\mathcal{F}$ is universally pure relative to $S$.

Proof. Immediate consequence of Lemma 38.18.3 and the definitions. $\square$

Lemma 38.18.5. Let $I$ be a directed set. Let $(S_ i, g_{ii'})$ be an inverse system of affine schemes over $I$. Set $S = \mathop{\mathrm{lim}}\nolimits _ i S_ i$ and $s \in S$. Denote $g_ i : S \to S_ i$ the projections and set $s_ i = g_ i(s)$. Suppose that $f : X \to S$ is a morphism of finite presentation, $\mathcal{F}$ a quasi-coherent $\mathcal{O}_ X$-module of finite presentation which is pure along $X_ s$ and flat over $S$ at all points of $X_ s$. Then there exists an $i \in I$, a morphism of finite presentation $X_ i \to S_ i$, a quasi-coherent $\mathcal{O}_{X_ i}$-module $\mathcal{F}_ i$ of finite presentation which is pure along $(X_ i)_{s_ i}$ and flat over $S_ i$ at all points of $(X_ i)_{s_ i}$ such that $X \cong X_ i \times _{S_ i} S$ and such that the pullback of $\mathcal{F}_ i$ to $X$ is isomorphic to $\mathcal{F}$.

Proof. Let $U \subset X$ be the set of points where $\mathcal{F}$ is flat over $S$. By More on Morphisms, Theorem 37.15.1 this is an open subscheme of $X$. By assumption $X_ s \subset U$. As $X_ s$ is quasi-compact, we can find a quasi-compact open $U' \subset U$ with $X_ s \subset U'$. By Limits, Lemma 32.10.1 we can find an $i \in I$ and a morphism of finite presentation $f_ i : X_ i \to S_ i$ whose base change to $S$ is isomorphic to $f_ i$. Fix such a choice and set $X_{i'} = X_ i \times _{S_ i} S_{i'}$. Then $X = \mathop{\mathrm{lim}}\nolimits _{i'} X_{i'}$ with affine transition morphisms. By Limits, Lemma 32.10.2 we can, after possible increasing $i$ assume there exists a quasi-coherent $\mathcal{O}_{X_ i}$-module $\mathcal{F}_ i$ of finite presentation whose base change to $S$ is isomorphic to $\mathcal{F}$. By Limits, Lemma 32.4.11 after possibly increasing $i$ we may assume there exists an open $U'_ i \subset X_ i$ whose inverse image in $X$ is $U'$. Note that in particular $(X_ i)_{s_ i} \subset U'_ i$. By Limits, Lemma 32.10.4 (after increasing $i$ once more) we may assume that $\mathcal{F}_ i$ is flat on $U'_ i$. In particular we see that $\mathcal{F}_ i$ is flat along $(X_ i)_{s_ i}$.

Next, we use Lemma 38.12.5 to choose an elementary étale neighbourhood $(S_ i', s_ i') \to (S_ i, s_ i)$ and a commutative diagram of schemes

\[ \xymatrix{ X_ i \ar[d] & X_ i' \ar[l]^{g_ i} \ar[d] \\ S_ i & S_ i' \ar[l] } \]

such that $g_ i$ is étale, $(X_ i)_{s_ i} \subset g_ i(X_ i')$, the schemes $X_ i'$, $S_ i'$ are affine, and such that $\Gamma (X_ i', g_ i^*\mathcal{F}_ i)$ is a projective $\Gamma (S_ i', \mathcal{O}_{S_ i'})$-module. Note that $g_ i^*\mathcal{F}_ i$ is universally pure over $S'_ i$, see Lemma 38.17.4. We may base change the diagram above to a diagram with morphisms $(S'_{i'}, s'_{i'}) \to (S_{i'}, s_{i'})$ and $g_{i'} : X'_{i'} \to X_{i'}$ over $S_{i'}$ for any $i' \geq i$ and we may base change the diagram to a diagram with morphisms $(S', s') \to (S, s)$ and $g : X' \to X$ over $S$.

At this point we can use our criterion for purity. Set $W'_ i \subset X_ i \times _{S_ i} S'_ i$ equal to the image of the étale morphism $X'_ i \to X_ i \times _{S_ i} S'_ i$. For every $i' \geq i$ we have similarly the image $W'_{i'} \subset X_{i'} \times _{S_{i'}} S'_{i'}$ and we have the image $W' \subset X \times _ S S'$. Taking images commutes with base change, hence $W'_{i'} = W'_ i \times _{S'_ i} S'_{i'}$ and $W' = W_ i \times _{S'_ i} S'$. Because $\mathcal{F}$ is pure along $X_ s$ the Lemma 38.18.2 implies that

38.18.5.1
\begin{equation} \label{flat-equation-inclusion} f^{-1}(\mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})) \cap \text{Ass}_{X \times _ S S'/S'}(\mathcal{F} \times _ S S') \subset W' \end{equation}

By More on Morphisms, Lemma 37.23.5 we see that

\[ E = \{ t \in S' \mid \text{Ass}_{X_ t}(\mathcal{F}_ t) \subset W' \} \quad \text{and}\quad E_{i'} = \{ t \in S'_{i'} \mid \text{Ass}_{X_ t}(\mathcal{F}_{i', t}) \subset W'_{i'} \} \]

are locally constructible subsets of $S'$ and $S'_{i'}$. By More on Morphisms, Lemma 37.23.4 we see that $E_{i'}$ is the inverse image of $E_ i$ under the morphism $S'_{i'} \to S'_ i$ and that $E$ is the inverse image of $E_ i$ under the morphism $S' \to S'_ i$. Thus Equation (38.18.5.1) is equivalent to the assertion that $\mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ maps into $E_ i$. As $\mathcal{O}_{S', s'} = \mathop{\mathrm{colim}}\nolimits _{i' \geq i} \mathcal{O}_{S'_{i'}, s'_{i'}}$ we see that $\mathop{\mathrm{Spec}}(\mathcal{O}_{S'_{i'}, s'_{i'}})$ maps into $E_ i$ for some $i' \geq i$, see Limits, Lemma 32.4.10. Then, applying Lemma 38.18.2 to the situation over $S_{i'}$, we conclude that $\mathcal{F}_{i'}$ is pure along $(X_{i'})_{s_{i'}}$. $\square$

Lemma 38.18.6. Let $f : X \to S$ be a morphism of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite presentation flat over $S$. Then the set

\[ U = \{ s \in S \mid \mathcal{F}\text{ is pure along }X_ s\} \]

is open in $S$.

Proof. Let $s \in U$. Using Lemma 38.12.5 we can find an elementary étale neighbourhood $(S', s') \to (S, s)$ and a commutative diagram

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

such that $g$ is étale, $X_ s \subset g(X')$, the schemes $X'$, $S'$ are affine, and such that $\Gamma (X', g^*\mathcal{F})$ is a projective $\Gamma (S', \mathcal{O}_{S'})$-module. Note that $g^*\mathcal{F}$ is universally pure over $S'$, see Lemma 38.17.4. Set $W' \subset X \times _ S S'$ equal to the image of the étale morphism $X' \to X \times _ S S'$. Note that $W$ is open and quasi-compact over $S'$. Set

\[ E = \{ t \in S' \mid \text{Ass}_{X_ t}(\mathcal{F}_ t) \subset W' \} . \]

By More on Morphisms, Lemma 37.23.5 $E$ is a constructible subset of $S'$. By Lemma 38.18.2 we see that $\mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \subset E$. By Morphisms, Lemma 29.22.4 we see that $E$ contains an open neighbourhood $V'$ of $s'$. Applying Lemma 38.18.2 once more we see that for any point $s_1$ in the image of $V'$ in $S$ the sheaf $\mathcal{F}$ is pure along $X_{s_1}$. Since $S' \to S$ is étale the image of $V'$ in $S$ is open and we win. $\square$


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