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 34.3.14 or 34.4.14. 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 34.3.13, 34.3.17, 34.4.13, and 34.4.17. 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 35.13.6 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. $\square$

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