The Stacks project

59.39 Comparing topologies

In this section we start studying what happens when you compare sheaves with respect to different topologies.

Lemma 59.39.1. Let $S$ be a scheme. Let $\mathcal{F}$ be a sheaf of sets on $S_{\acute{e}tale}$. Let $s, t \in \mathcal{F}(S)$. Then there exists an open $W \subset S$ characterized by the following property: A morphism $f : T \to S$ factors through $W$ if and only if $s|_ T = t|_ T$ (restriction is pullback by $f_{small}$).

Proof. Consider the presheaf which assigns to $U \in \mathop{\mathrm{Ob}}\nolimits (S_{\acute{e}tale})$ the empty set if $s|_ U \not= t|_ U$ and a singleton else. It is clear that this is a subsheaf of the final object of $\mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale})$. By Lemma 59.31.1 we find an open $W \subset S$ representing this presheaf. For a geometric point $\overline{x}$ of $S$ we see that $\overline{x} \in W$ if and only if the stalks of $s$ and $t$ at $\overline{x}$ agree. By the description of stalks of pullbacks in Lemma 59.36.2 we see that $W$ has the desired property. $\square$

Lemma 59.39.2. Let $S$ be a scheme. Let $\tau \in \{ Zariski, {\acute{e}tale}\} $. Consider the morphism

\[ \pi _ S : (\mathit{Sch}/S)_\tau \longrightarrow S_\tau \]

of Topologies, Lemma 34.3.13 or 34.4.13. Let $\mathcal{F}$ be a sheaf on $S_\tau $. Then $\pi _ S^{-1}\mathcal{F}$ is given by the rule

\[ (\pi _ S^{-1}\mathcal{F})(T) = \Gamma (T_\tau , f_{small}^{-1}\mathcal{F}) \]

where $f : T \to S$. Moreover, $\pi _ S^{-1}\mathcal{F}$ satisfies the sheaf condition with respect to fpqc coverings.

Proof. Observe that we have a morphism $i_ f : \mathop{\mathit{Sh}}\nolimits (T_\tau ) \to \mathop{\mathit{Sh}}\nolimits (\mathit{Sch}/S)_\tau )$ such that $\pi _ S \circ i_ f = f_{small}$ as morphisms $T_\tau \to S_\tau $, see Topologies, Lemmas 34.3.12, 34.3.16, 34.4.12, and 34.4.16. Since pullback is transitive we see that $i_ f^{-1} \pi _ S^{-1}\mathcal{F} = f_{small}^{-1}\mathcal{F}$ as desired.

Let $\{ g_ i : T_ i \to T\} _{i \in I}$ be an fpqc covering. The final statement means the following: Given a sheaf $\mathcal{G}$ on $T_\tau $ and given sections $s_ i \in \Gamma (T_ i, g_{i, small}^{-1}\mathcal{G})$ whose pullbacks to $T_ i \times _ T T_ j$ agree, there is a unique section $s$ of $\mathcal{G}$ over $T$ whose pullback to $T_ i$ agrees with $s_ i$.

Let $V \to T$ be an object of $T_\tau $ and let $t \in \mathcal{G}(V)$. For every $i$ there is a largest open $W_ i \subset T_ i \times _ T V$ such that the pullbacks of $s_ i$ and $t$ agree as sections of the pullback of $\mathcal{G}$ to $W_ i \subset T_ i \times _ T V$, see Lemma 59.39.1. Because $s_ i$ and $s_ j$ agree over $T_ i \times _ T T_ j$ we find that $W_ i$ and $W_ j$ pullback to the same open over $T_ i \times _ T T_ j \times _ T V$. By Descent, Lemma 35.12.6 we find an open $W \subset V$ whose inverse image to $T_ i \times _ T V$ recovers $W_ i$.

By construction of $g_{i, small}^{-1}\mathcal{G}$ there exists a $\tau $-covering $\{ T_{ij} \to T_ i\} _{j \in J_ i}$, for each $j$ an open immersion or étale morphism $V_{ij} \to T$, a section $t_{ij} \in \mathcal{G}(V_{ij})$, and commutative diagrams

\[ \xymatrix{ T_{ij} \ar[r] \ar[d] & V_{ij} \ar[d] \\ T_ i \ar[r] & T } \]

such that $s_ i|_{T_{ij}}$ is the pullback of $t_{ij}$. In other words, after replacing the covering $\{ T_ i \to T\} $ by $\{ T_{ij} \to T\} $ we may assume there are factorizations $T_ i \to V_ i \to T$ with $V_ i \in \mathop{\mathrm{Ob}}\nolimits (T_\tau )$ and sections $t_ i \in \mathcal{G}(V_ i)$ pulling back to $s_ i$ over $T_ i$. By the result of the previous paragraph we find opens $W_ i \subset V_ i$ such that $t_ i|_{W_ i}$ “agrees with” every $s_ j$ over $T_ j \times _ T W_ i$. Note that $T_ i \to V_ i$ factors through $W_ i$. Hence $\{ W_ i \to T\} $ is a $\tau $-covering and the lemma is proven. $\square$

Lemma 59.39.3. Let $S$ be a scheme. Let $f : T \to S$ be a morphism such that

  1. $f$ is flat and quasi-compact, and

  2. the geometric fibres of $f$ are connected.

Let $\mathcal{F}$ be a sheaf on $S_{\acute{e}tale}$. Then $\Gamma (S, \mathcal{F}) = \Gamma (T, f^{-1}_{small}\mathcal{F})$.

Proof. There is a canonical map $\Gamma (S, \mathcal{F}) \to \Gamma (T, f_{small}^{-1}\mathcal{F})$. Since $f$ is surjective (because its fibres are connected) we see that this map is injective.

To show that the map is surjective, let $\alpha \in \Gamma (T, f_{small}^{-1}\mathcal{F})$. Since $\{ T \to S\} $ is an fpqc covering we can use Lemma 59.39.2 to see that suffices to prove that $\alpha $ pulls back to the same section over $T \times _ S T$ by the two projections. Let $\overline{s} \to S$ be a geometric point. It suffices to show the agreement holds over $(T \times _ S T)_{\overline{s}}$ as every geometric point of $T \times _ S T$ is contained in one of these geometric fibres. In other words, we are trying to show that $\alpha |_{T_{\overline{s}}}$ pulls back to the same section over

\[ (T \times _ S T)_{\overline{s}} = T_{\overline{s}} \times _{\overline{s}} T_{\overline{s}} \]

by the two projections to $T_{\overline{s}}$. However, since $\mathcal{F}|_{T_{\overline{s}}}$ is the pullback of $\mathcal{F}|_{\overline{s}}$ it is a constant sheaf with value $\mathcal{F}_{\overline{s}}$. Since $T_{\overline{s}}$ is connected by assumption, any section of a constant sheaf is constant. Hence $\alpha |_{T_{\overline{s}}}$ corresponds to an element of $\mathcal{F}_{\overline{s}}$. Thus the two pullbacks to $(T \times _ S T)_{\overline{s}}$ both correspond to this same element and we conclude. $\square$

Here is a version of Lemma 59.39.3 where we do not assume that the morphism is flat.

Lemma 59.39.4. Let $S$ be a scheme. Let $f : X \to S$ be a morphism such that

  1. $f$ is submersive, and

  2. the geometric fibres of $f$ are connected.

Let $\mathcal{F}$ be a sheaf on $S_{\acute{e}tale}$. Then $\Gamma (S, \mathcal{F}) = \Gamma (X, f^{-1}_{small}\mathcal{F})$.

Proof. There is a canonical map $\Gamma (S, \mathcal{F}) \to \Gamma (X, f_{small}^{-1}\mathcal{F})$. Since $f$ is surjective (because its fibres are connected) we see that this map is injective.

To show that the map is surjective, let $\tau \in \Gamma (X, f_{small}^{-1}\mathcal{F})$. It suffices to find an étale covering $\{ U_ i \to S\} $ and sections $\sigma _ i \in \mathcal{F}(U_ i)$ such that $\sigma _ i$ pulls back to $\tau |_{X \times _ S U_ i}$. Namely, the injectivity shown above guarantees that $\sigma _ i$ and $\sigma _ j$ restrict to the same section of $\mathcal{F}$ over $U_ i \times _ S U_ j$. Thus we obtain a unique section $\sigma \in \mathcal{F}(S)$ which restricts to $\sigma _ i$ over $U_ i$. Then the pullback of $\sigma $ to $X$ is $\tau $ because this is true locally.

Let $\overline{x}$ be a geometric point of $X$ with image $\overline{s}$ in $S$. Consider the image of $\tau $ in the stalk

\[ (f_{small}^{-1}\mathcal{F})_{\overline{x}} = \mathcal{F}_{\overline{s}} \]

See Lemma 59.36.2. We can find an étale neighbourhood $U \to S$ of $\overline{s}$ and a section $\sigma \in \mathcal{F}(U)$ mapping to this image in the stalk. Thus after replacing $S$ by $U$ and $X$ by $X \times _ S U$ we may assume there exits a section $\sigma $ of $\mathcal{F}$ over $S$ whose image in $(f_{small}^{-1}\mathcal{F})_{\overline{x}}$ is the same as $\tau $.

By Lemma 59.39.1 there exists a maximal open $W \subset X$ such that $f_{small}^{-1}\sigma $ and $\tau $ agree over $W$ and the formation of $W$ commutes with further pullback. Observe that the pullback of $\mathcal{F}$ to the geometric fibre $X_{\overline{s}}$ is the pullback of $\mathcal{F}_{\overline{s}}$ viewed as a sheaf on $\overline{s}$ by $X_{\overline{s}} \to \overline{s}$. Hence we see that $\tau $ and $\sigma $ give sections of the constant sheaf with value $\mathcal{F}_{\overline{s}}$ on $X_{\overline{s}}$ which agree in one point. Since $X_{\overline{s}}$ is connected by assumption, we conclude that $W$ contains $X_ s$. The same argument for different geometric fibres shows that $W$ contains every fibre it meets. Since $f$ is submersive, we conclude that $W$ is the inverse image of an open neighbourhood of $s$ in $S$. This finishes the proof. $\square$

Lemma 59.39.5. Let $K/k$ be an extension of fields with $k$ separably algebraically closed. Let $S$ be a scheme over $k$. Denote $p : S_ K = S \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(K) \to S$ the projection. Let $\mathcal{F}$ be a sheaf on $S_{\acute{e}tale}$. Then $\Gamma (S, \mathcal{F}) = \Gamma (S_ K, p^{-1}_{small}\mathcal{F})$.

Proof. Follows from Lemma 59.39.3. Namely, it is clear that $p$ is flat and quasi-compact as the base change of $\mathop{\mathrm{Spec}}(K) \to \mathop{\mathrm{Spec}}(k)$. On the other hand, if $\overline{s} : \mathop{\mathrm{Spec}}(L) \to S$ is a geometric point, then the fibre of $p$ over $\overline{s}$ is the spectrum of $K \otimes _ k L$ which is irreducible hence connected by Algebra, Lemma 10.47.2. $\square$

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 09XL. Beware of the difference between the letter 'O' and the digit '0'.