The Stacks project

38.2 Lemmas on étale localization

In this section we list some lemmas on étale localization which will be useful later in this chapter. Please skip this section on a first reading.

Lemma 38.2.1. Let $i : Z \to X$ be a closed immersion of affine schemes. Let $Z' \to Z$ be an étale morphism with $Z'$ affine. Then there exists an étale morphism $X' \to X$ with $X'$ affine such that $Z' \cong Z \times _ X X'$ as schemes over $Z$.

Proof. See Algebra, Lemma 10.143.10. $\square$

Lemma 38.2.2. Let

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

be a commutative diagram of schemes with $X' \to X$ and $S' \to S$ étale. Let $s' \in S'$ be a point. Then

\[ X' \times _{S'} \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \longrightarrow X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \]

is étale.

Proof. This is true because $X' \to X_{S'}$ is étale as a morphism of schemes étale over $X$, see Morphisms, Lemma 29.36.18 and the base change of an étale morphism is étale, see Morphisms, Lemma 29.36.4. $\square$

Lemma 38.2.3. Let $X \to T \to S$ be morphisms of schemes with $T \to S$ étale. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $x \in X$ be a point. Then

\[ \mathcal{F}\text{ flat over }S\text{ at }x \Leftrightarrow \mathcal{F}\text{ flat over }T\text{ at }x \]

In particular $\mathcal{F}$ is flat over $S$ if and only if $\mathcal{F}$ is flat over $T$.

Proof. As an étale morphism is a flat morphism (see Morphisms, Lemma 29.36.12) the implication “$\Leftarrow $” follows from Algebra, Lemma 10.39.4. For the converse assume that $\mathcal{F}$ is flat at $x$ over $S$. Denote $\tilde x \in X \times _ S T$ the point lying over $x$ in $X$ and over the image of $x$ in $T$ in $T$. Then $(X \times _ S T \to X)^*\mathcal{F}$ is flat at $\tilde x$ over $T$ via $\text{pr}_2 : X \times _ S T \to T$, see Morphisms, Lemma 29.25.7. The diagonal $\Delta _{T/S} : T \to T \times _ S T$ is an open immersion; combine Morphisms, Lemmas 29.35.13 and 29.36.5. So $X$ is identified with open subscheme of $X \times _ S T$, the restriction of $\text{pr}_2$ to this open is the given morphism $X \to T$, the point $\tilde x$ corresponds to the point $x$ in this open, and $(X \times _ S T \to X)^*\mathcal{F}$ restricted to this open is $\mathcal{F}$. Whence we see that $\mathcal{F}$ is flat at $x$ over $T$. $\square$

Lemma 38.2.4. Let $T \to S$ be an étale morphism. Let $t \in T$ with image $s \in S$. Let $M$ be a $\mathcal{O}_{T, t}$-module. Then

\[ M\text{ flat over }\mathcal{O}_{S, s} \Leftrightarrow M\text{ flat over }\mathcal{O}_{T, t}. \]

Proof. We may replace $S$ by an affine neighbourhood of $s$ and after that $T$ by an affine neighbourhood of $t$. Set $\mathcal{F} = (\mathop{\mathrm{Spec}}(\mathcal{O}_{T, t}) \to T)_*\widetilde M$. This is a quasi-coherent sheaf (see Schemes, Lemma 26.24.1 or argue directly) on $T$ whose stalk at $t$ is $M$ (details omitted). Apply Lemma 38.2.3. $\square$

Lemma 38.2.5. Let $S$ be a scheme and $s \in S$ a point. Denote $\mathcal{O}_{S, s}^ h$ (resp. $\mathcal{O}_{S, s}^{sh}$) the henselization (resp. strict henselization), see Algebra, Definition 10.155.3. Let $M^{sh}$ be a $\mathcal{O}_{S, s}^{sh}$-module. The following are equivalent

  1. $M^{sh}$ is flat over $\mathcal{O}_{S, s}$,

  2. $M^{sh}$ is flat over $\mathcal{O}_{S, s}^ h$, and

  3. $M^{sh}$ is flat over $\mathcal{O}_{S, s}^{sh}$.

If $M^{sh} = M^ h \otimes _{\mathcal{O}_{S, s}^ h} \mathcal{O}_{S, s}^{sh}$ this is also equivalent to

  1. $M^ h$ is flat over $\mathcal{O}_{S, s}$, and

  2. $M^ h$ is flat over $\mathcal{O}_{S, s}^ h$.

If $M^ h = M \otimes _{\mathcal{O}_{S, s}} \mathcal{O}_{S, s}^ h$ this is also equivalent to

  1. $M$ is flat over $\mathcal{O}_{S, s}$.

Proof. By More on Algebra, Lemma 15.45.1 the local ring maps $\mathcal{O}_{S, s} \to \mathcal{O}_{S, s}^ h \to \mathcal{O}_{S, s}^{sh}$ are faithfully flat. Hence (3) $\Rightarrow $ (2) $\Rightarrow $ (1) and (5) $\Rightarrow $ (4) follow from Algebra, Lemma 10.39.4. By faithful flatness the equivalences (6) $\Leftrightarrow $ (5) and (5) $\Leftrightarrow $ (3) follow from Algebra, Lemma 10.39.8. Thus it suffices to show that (1) $\Rightarrow $ (2) $\Rightarrow $ (3) and (4) $\Rightarrow $ (5). To prove these we may assume $S$ is an affine scheme.

Assume (1). By Lemma 38.2.4 we see that $M^{sh}$ is flat over $\mathcal{O}_{T, t}$ for any étale neighbourhood $(T, t) \to (S, s)$. Since $\mathcal{O}_{S, s}^ h$ and $\mathcal{O}_{S, s}^{sh}$ are directed colimits of local rings of the form $\mathcal{O}_{T, t}$ (see Algebra, Lemmas 10.155.7 and 10.155.11) we conclude that $M^{sh}$ is flat over $\mathcal{O}_{S, s}^ h$ and $\mathcal{O}_{S, s}^{sh}$ by Algebra, Lemma 10.39.6. Thus (1) implies (2) and (3). Of course this implies also (2) $\Rightarrow $ (3) by replacing $\mathcal{O}_{S, s}$ by $\mathcal{O}_{S, s}^ h$. The same argument applies to prove (4) $\Rightarrow $ (5). $\square$

Lemma 38.2.6. Let $S$ be a scheme and $s \in S$ a point. Denote $\mathcal{O}_{S, s}^ h$ (resp. $\mathcal{O}_{S, s}^{sh}$) the henselization (resp. strict henselization), see Algebra, Definition 10.155.3. Let $M^{sh}$ be an object of $D(\mathcal{O}_{S, s}^{sh})$. Let $a, b \in \mathbf{Z}$. The following are equivalent

  1. $M^{sh}$ has tor amplitude in $[a, b]$ over $\mathcal{O}_{S, s}$,

  2. $M^{sh}$ has tor amplitude in $[a, b]$ over $\mathcal{O}_{S, s}^ h$, and

  3. $M^{sh}$ has tor amplitude in $[a, b]$ over $\mathcal{O}_{S, s}^{sh}$.

If $M^{sh} = M^ h \otimes _{\mathcal{O}_{S, s}^ h}^\mathbf {L} \mathcal{O}_{S, s}^{sh}$ for $M^ h \in D(\mathcal{O}_{S, s}^ h)$ this is also equivalent to

  1. $M^ h$ has tor amplitude in $[a, b]$ over $\mathcal{O}_{S, s}$, and

  2. $M^ h$ has tor amplitude in $[a, b]$ over $\mathcal{O}_{S, s}^ h$.

If $M^ h = M \otimes _{\mathcal{O}_{S, s}}^\mathbf {L} \mathcal{O}_{S, s}^ h$ for $M \in D(\mathcal{O}_{S, s})$ this is also equivalent to

  1. $M$ has tor amplitude in $[a, b]$ over $\mathcal{O}_{S, s}$.

Proof. By More on Algebra, Lemma 15.45.1 the local ring maps $\mathcal{O}_{S, s} \to \mathcal{O}_{S, s}^ h \to \mathcal{O}_{S, s}^{sh}$ are faithfully flat. Hence (3) $\Rightarrow $ (2) $\Rightarrow $ (1) and (5) $\Rightarrow $ (4) follow from More on Algebra, Lemma 15.66.11. By faithful flatness the equivalences (6) $\Leftrightarrow $ (5) and (5) $\Leftrightarrow $ (3) follow from More on Algebra, Lemma 15.66.17. Thus it suffices to show that (1) $\Rightarrow $ (3), (2) $\Rightarrow $ (3), and (4) $\Rightarrow $ (5).

Assume (1). In particular $M^{sh}$ has vanishing cohomology in degrees $< a$ and $> b$. Hence we can represent $M^{sh}$ by a complex $P^\bullet $ of free $\mathcal{O}_{X, x}^{sh}$-modules with $P^ i = 0$ for $i > b$ (see for example the very general Derived Categories, Lemma 13.15.4). Note that $P^ n$ is flat over $\mathcal{O}_{S, s}$ for all $n$. Consider $\mathop{\mathrm{Coker}}(d_ P^{a - 1})$. By More on Algebra, Lemma 15.66.2 this is a flat $\mathcal{O}_{S, s}$-module. Hence by Lemma 38.2.5 this is a flat $\mathcal{O}_{S, s}^{sh}$-module. Thus $\tau _{\geq a}P^\bullet $ is a complex of flat $\mathcal{O}_{S, s}^{sh}$-modules representing $M^{sh}$ in $D(\mathcal{O}_{S, s}^{sh}$ and we find that $M^{sh}$ has tor amplitude in $[a, b]$, see More on Algebra, Lemma 15.66.3. Thus (1) implies (3). Of course this implies also (2) $\Rightarrow $ (3) by replacing $\mathcal{O}_{S, s}$ by $\mathcal{O}_{S, s}^ h$. The same argument applies to prove (4) $\Rightarrow $ (5). $\square$

Lemma 38.2.7. Let $g : T \to S$ be a finite flat morphism of schemes. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ S$-module. Let $t \in T$ be a point with image $s \in S$. Then

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

Proof. The implication “$\Leftarrow $” follows immediately from Divisors, Lemma 31.6.4. Assume $t \in \text{WeakAss}(g^*\mathcal{G})$. Let $\mathop{\mathrm{Spec}}(A) \subset S$ be an affine open neighbourhood of $s$. Let $\mathcal{G}$ be the quasi-coherent sheaf associated to the $A$-module $M$. Let $\mathfrak p \subset A$ be the prime ideal corresponding to $s$. As $g$ is finite flat we have $g^{-1}(\mathop{\mathrm{Spec}}(A)) = \mathop{\mathrm{Spec}}(B)$ for some finite flat $A$-algebra $B$. Note that $g^*\mathcal{G}$ is the quasi-coherent $\mathcal{O}_{\mathop{\mathrm{Spec}}(B)}$-module associated to the $B$-module $M \otimes _ A B$ and $g_*g^*\mathcal{G}$ is the quasi-coherent $\mathcal{O}_{\mathop{\mathrm{Spec}}(A)}$-module associated to the $A$-module $M \otimes _ A B$. By Algebra, Lemma 10.78.5 we have $B_{\mathfrak p} \cong A_{\mathfrak p}^{\oplus n}$ for some integer $n \geq 0$. Note that $n \geq 1$ as we assumed there exists at least one point of $T$ lying over $s$. Hence we see by looking at stalks that

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

Now the assumption that $t \in \text{WeakAss}(g^*\mathcal{G})$ implies that $s \in \text{WeakAss}(g_*g^*\mathcal{G})$ by Divisors, Lemma 31.6.3 and hence by the above $s \in \text{WeakAss}(\mathcal{G})$. $\square$

Lemma 38.2.8. Let $h : U \to S$ be an étale morphism of schemes. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ S$-module. Let $u \in U$ be a point with image $s \in S$. Then

\[ u \in \text{WeakAss}(h^*\mathcal{G}) \Leftrightarrow s \in \text{WeakAss}(\mathcal{G}) \]

Proof. After replacing $S$ and $U$ by affine neighbourhoods of $s$ and $u$ we may assume that $g$ is a standard étale morphism of affines, see Morphisms, Lemma 29.36.14. Thus we may assume $S = \mathop{\mathrm{Spec}}(A)$ and $X = \mathop{\mathrm{Spec}}(A[x, 1/g]/(f))$, where $f$ is monic and $f'$ is invertible in $A[x, 1/g]$. Note that $A[x, 1/g]/(f) = (A[x]/(f))_ g$ is also the localization of the finite free $A$-algebra $A[x]/(f)$. Hence we may think of $U$ as an open subscheme of the scheme $T = \mathop{\mathrm{Spec}}(A[x]/(f))$ which is finite locally free over $S$. This reduces us to Lemma 38.2.7 above. $\square$

Lemma 38.2.9. Let $S$ be a scheme and $s \in S$ a point. Denote $\mathcal{O}_{S, s}^ h$ (resp. $\mathcal{O}_{S, s}^{sh}$) the henselization (resp. strict henselization), see Algebra, Definition 10.155.3. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ S$-module. The following are equivalent

  1. $s$ is a weakly associated point of $\mathcal{F}$,

  2. $\mathfrak m_ s$ is a weakly associated prime of $\mathcal{F}_ s$,

  3. $\mathfrak m_ s^ h$ is a weakly associated prime of $\mathcal{F}_ s \otimes _{\mathcal{O}_{S, s}} \mathcal{O}_{S, s}^ h$, and

  4. $\mathfrak m_ s^{sh}$ is a weakly associated prime of $\mathcal{F}_ s \otimes _{\mathcal{O}_{S, s}} \mathcal{O}_{S, s}^{sh}$.

Proof. The equivalence of (1) and (2) is the definition, see Divisors, Definition 31.5.1. The implications (2) $\Rightarrow $ (3) $\Rightarrow $ (4) follows from Divisors, Lemma 31.6.4 applied to the flat (More on Algebra, Lemma 15.45.1) morphisms

\[ \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^{sh}) \to \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^ h) \to \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) \]

and the closed points. To prove (4) $\Rightarrow $ (2) we may replace $S$ by an affine neighbourhood. Suppose that $x \in \mathcal{F}_ s \otimes _{\mathcal{O}_{S, s}} \mathcal{O}_{S, s}^{sh}$ is an element whose annihilator has radical equal to $\mathfrak m_ s^{sh}$. (See Algebra, Lemma 10.66.2.) Since $\mathcal{O}_{S, s}^{sh}$ is equal to the limit of $\mathcal{O}_{U, u}$ over étale neighbourhoods $f : (U, u) \to (S, s)$ by Algebra, Lemma 10.155.11 we may assume that $x$ is the image of some $x' \in \mathcal{F}_ s \otimes _{\mathcal{O}_{S, s}} \mathcal{O}_{U, u}$. The local ring map $\mathcal{O}_{U, u} \to \mathcal{O}_{S, s}^{sh}$ is faithfully flat (as it is the strict henselization), hence universally injective (Algebra, Lemma 10.82.11). It follows that the annihilator of $x'$ is the inverse image of the annihilator of $x$. Hence the radical of this annihilator is equal to $\mathfrak m_ u$. Thus $u$ is a weakly associated point of $f^*\mathcal{F}$. By Lemma 38.2.8 we see that $s$ is a weakly associated point of $\mathcal{F}$. $\square$


Comments (0)


Post a comment

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05FM. Beware of the difference between the letter 'O' and the digit '0'.