The Stacks project

58.103 Descending étale sheaves

We prove that étale sheaves “glue” in the fppf and h topology and related results. We have already shown the following related results

  1. Lemma 58.39.2 tells us that a sheaf on the small étale site of a scheme $S$ determines a sheaf on the big étale site of $S$ satisfying the sheaf condition for fpqc coverings (and a fortiori for Zariski, étale, smooth, syntomic, and fppf coverings),

  2. Lemma 58.99.1 is a restatement of the previous point for the fppf topology,

  3. Lemma 58.101.1 proves the same for the ph topology,

  4. Lemma 58.102.1 proves the same for the h topology,

  5. Lemma 58.99.4 is a version of fppf descent for étale sheaves, and

  6. Remark 58.55.6 tells us that we have descent of étale sheaves for finite surjective morphisms (we will clarify and strengthen this below).

In the chapter on simplicial spaces we will prove some additional results on this, see for example Simplicial Spaces, Sections 83.33 and 83.36.

In order to conveniently express our results we need some notation. Let $\mathcal{U} = \{ f_ i : X_ i \to X\} $ be a family of morphisms of schemes with fixed target. A descent datum for étale sheaves with respect to $\mathcal{U}$ is a family $((\mathcal{F}_ i)_{i \in I}, (\varphi _{ij})_{i, j \in I})$ where

  1. $\mathcal{F}_ i$ is in $\mathop{\mathit{Sh}}\nolimits (X_{i, {\acute{e}tale}})$, and

  2. $\varphi _{ij} : \text{pr}_{0, small}^{-1} \mathcal{F}_ i \longrightarrow \text{pr}_{1, small}^{-1} \mathcal{F}_ j$ is an isomorphism in $\mathop{\mathit{Sh}}\nolimits ((X_ i \times _ X X_ j)_{\acute{e}tale})$

such that the cocycle condition holds: the diagrams

\[ \xymatrix{ \text{pr}_{0, small}^{-1}\mathcal{F}_ i \ar[dr]_{\text{pr}_{02, small}^{-1}\varphi _{ik}} \ar[rr]^{\text{pr}_{01, small}^{-1}\varphi _{ij}} & & \text{pr}_{1, small}^{-1}\mathcal{F}_ j \ar[dl]^{\text{pr}_{12, small}^{-1}\varphi _{jk}} \\ & \text{pr}_{2, small}^{-1}\mathcal{F}_ k } \]

commute in $\mathop{\mathit{Sh}}\nolimits ((X_ i \times _ X X_ j \times _ X X_ k)_{\acute{e}tale})$. There is an obvious notion of morphisms of descent data and we obtain a category of descent data. A descent datum $((\mathcal{F}_ i)_{i \in I}, (\varphi _{ij})_{i, j \in I})$ is called effective if there exist a $\mathcal{F}$ in $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ and isomorphisms $\varphi _ i : f_{i, small}^{-1} \mathcal{F} \to \mathcal{F}_ i$ in $\mathop{\mathit{Sh}}\nolimits (X_{i, {\acute{e}tale}})$ compatible with the $\varphi _{ij}$, i.e., such that

\[ \varphi _{ij} = \text{pr}_{1, small}^{-1} (\varphi _ j) \circ \text{pr}_{0, small}^{-1} (\varphi _ i^{-1}) \]

Another way to say this is the following. Given an object $\mathcal{F}$ of $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ we obtain the canonical descent datum $(f_{i, small}^{-1}\mathcal{F}_ i, c_{ij})$ where $c_{ij}$ is the canonical isomorphism

\[ c_{ij} : \text{pr}_{0, small}^{-1} f_{i, small}^{-1}\mathcal{F} \longrightarrow \text{pr}_{1, small}^{-1} f_{j, small}^{-1}\mathcal{F} \]

The descent datum $((\mathcal{F}_ i)_{i \in I}, (\varphi _{ij})_{i, j \in I})$ is effective if and only if it is isomorphic to the canonical descent datum associated to some $\mathcal{F}$ in $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$.

If the family consists of a single morphism $\{ X \to Y\} $, then we think of a descent datum as a pair $(\mathcal{F}, \varphi )$ where $\mathcal{F}$ is an object of $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ and $\varphi $ is an isomorphism

\[ \text{pr}_{0, small}^{-1} \mathcal{F} \longrightarrow \text{pr}_{1, small}^{-1} \mathcal{F} \]

in $\mathop{\mathit{Sh}}\nolimits ((X \times _ Y X)_{\acute{e}tale})$ such that the cocycle condition holds:

\[ \xymatrix{ \text{pr}_{0, small}^{-1}\mathcal{F} \ar[dr]_{\text{pr}_{02, small}^{-1}\varphi } \ar[rr]^{\text{pr}_{01, small}^{-1}\varphi } & & \text{pr}_{1, small}^{-1}\mathcal{F} \ar[dl]^{\text{pr}_{12, small}^{-1}\varphi } \\ & \text{pr}_{2, small}^{-1}\mathcal{F} } \]

commutes in $\mathop{\mathit{Sh}}\nolimits ((X \times _ Y X \times _ Y X)_{\acute{e}tale})$. There is a notion of morphisms of descent data and effectivity exactly as before.

We first prove effective descent for surjective integral morphisms.

Lemma 58.103.1. Let $f : X \to Y$ be a morphism of schemes which has a section. Then the functor

\[ \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt }\{ X \to Y\} \]

sending $\mathcal{G}$ in $\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$ to the canonical descent datum is an equivalence of categories.

Proof. This is formal and depends only on functoriality of the pullback functors. We omit the details. Hint: If $s : Y \to X$ is a section, then a quasi-inverse is the functor sending $(\mathcal{F}, \varphi )$ to $s_{small}^{-1}\mathcal{F}$. $\square$

Lemma 58.103.2. Let $f : X \to Y$ be a surjective integral morphism of schemes. The functor

\[ \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt }\{ X \to Y\} \]

is an equivalence of categories.

Proof. In this proof we drop the subscript ${}_{small}$ from our pullback and pushforward functors. Denote $X_1 = X \times _ Y X$ and denote $f_1 : X_1 \to Y$ the morphism $f \circ \text{pr}_0 = f \circ \text{pr}_1$. Let $(\mathcal{F}, \varphi )$ be a descent datum for $\{ X \to Y\} $. Let us set $\mathcal{F}_1 = \text{pr}_0^{-1}\mathcal{F}$. We may think of $\varphi $ as defining an isomorphism $\mathcal{F}_1 \to \text{pr}_1^{-1}\mathcal{F}$. We claim that the rule which sends a descent datum $(\mathcal{F}, \varphi )$ to the sheaf

\[ \mathcal{G} = \text{Equalizer}\left( \xymatrix{ f_*\mathcal{F} \ar@<1ex>[r] \ar@<-1ex>[r] & f_{1, *}\mathcal{F}_1 } \right) \]

is a quasi-inverse to the functor in the statement of the lemma. The first of the two arrows comes from the map

\[ f_*\mathcal{F} \to f_*\text{pr}_{0, *}\text{pr}_0^{-1}\mathcal{F} = f_{1, *}\mathcal{F}_1 \]

and the second arrow comes from the map

\[ f_*\mathcal{F} \to f_* \text{pr}_{1, *}\text{pr}_1^{-1}\mathcal{F} \xleftarrow {\varphi } f_* \text{pr}_{0, *} \text{pr}_0^{-1}\mathcal{F} = f_{1, *}\mathcal{F}_1 \]

where the arrow pointing left is invertible. To prove this works we have to show that the canonical map $f^{-1}\mathcal{G} \to \mathcal{F}$ is an isomorphism; details omitted. In order to prove this it suffices to check after pulling back by any collection of morphisms $\mathop{\mathrm{Spec}}(k) \to Y$ where $k$ is an algebraically closed field. Namely, the corresponing base changes $X_ k \to X$ are jointly surjective and we can check whether a map of sheaves on $X_{\acute{e}tale}$ is an isomorphism by looking at stalks on geometric points, see Theorem 58.29.10. By Lemma 58.55.4 the construction of $\mathcal{G}$ from the descent datum $(\mathcal{F}, \varphi )$ commutes with any base change. Thus we may assume $Y$ is the spectrum of an algebraically closed point (note that base change preserves the properties of the morphism $f$, see Morphisms, Lemma 29.9.4 and 29.44.6). In this case the morphism $X \to Y$ has a section, so we know that the functor is an equivalence by Lemma 58.103.1. However, the reader may show that the functor is an equivalence if and only if the construction above is a quasi-inverse; details omitted. This finishes the proof. $\square$

Lemma 58.103.3. Let $f : X \to Y$ be a surjective proper morphism of schemes. The functor

\[ \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt }\{ X \to Y\} \]

is an equivalence of categories.

Proof. The exact same proof as given in Lemma 58.103.2 works, except the appeal to Lemma 58.55.4 should be replaced by an appeal to Lemma 58.90.5. $\square$

Lemma 58.103.4. Let $f : X \to Y$ be a morphism of schemes. Let $Z \to Y$ be a surjective integral morphism of schemes or a surjective proper morphism of schemes. If the functors

\[ \mathop{\mathit{Sh}}\nolimits (Z_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt }\{ X \times _ Y Z \to Z\} \]

and

\[ \mathop{\mathit{Sh}}\nolimits ((Z \times _ Y Z)_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt } \{ X \times _ Y (Z \times _ Y Z) \to Z \times _ Y Z\} \]

are equivalences of categories, then

\[ \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt }\{ X \to Y\} \]

is an equivalence.

Proof. Formal consequence of the definitions and Lemmas 58.103.2 and 58.103.3. Details omitted. $\square$

Lemma 58.103.5. Let $f : X \to Y$ be a morphism of schemes which is surjective, flat, locally of finite presentation. The functor

\[ \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt }\{ X \to Y\} \]

is an equivalence of categories.

Proof. Exactly as in the proof of Lemma 58.103.2 we claim a quasi-inverse is given by the functor sending $(\mathcal{F}, \varphi )$ to

\[ \mathcal{G} = \text{Equalizer}\left( \xymatrix{ f_*\mathcal{F} \ar@<1ex>[r] \ar@<-1ex>[r] & f_{1, *}\mathcal{F}_1 } \right) \]

and in order to prove this it suffices to show that $f^{-1}\mathcal{G} \to \mathcal{F}$ is an isomorphism. This we may check locally, hence we may and do assume $Y$ is affine. Then we can find a finite surjective morphism $Z \to Y$ such that there exists an open covering $Z = \bigcup W_ i$ such that $W_ i \to Y$ factors through $X$. See More on Morphisms, Lemma 37.44.6. Applying Lemma 58.103.4 we see that it suffices to prove the lemma after replacing $Y$ by $Z$ and $Z \times _ Y Z$ and $f$ by its base change. Thus we may assume $f$ has sections Zariski locally. Of course, using that the problem is local on $Y$ we reduce to the case where we have a section which is Lemma 58.103.1. $\square$

Lemma 58.103.6. Let $\{ f_ i : X_ i \to X\} $ be an fppf covering of schemes. The functor

\[ \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt }\{ f_ i : X_ i \to X\} \]

is an equivalence of categories.

Proof. We have Lemma 58.103.5 for the morphism $f : \coprod X_ i \to X$. Then a formal argument shows that descent data for $f$ are the same thing as descent data for the covering, compare with Descent, Lemma 35.31.5. Details omitted. $\square$

Lemma 58.103.7. Let $f : X' \to X$ be a proper morphism of schemes. Let $i : Z \to X$ be a closed immersion. Set $E = Z \times _ X X'$. Picture

\[ \xymatrix{ E \ar[d]_ g \ar[r]_ j & X' \ar[d]^ f \\ Z \ar[r]^ i & X } \]

If $f$ is an isomorphism over $X \setminus Z$, then the functor

\[ \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (X'_{\acute{e}tale}) \times _{\mathop{\mathit{Sh}}\nolimits (E_{\acute{e}tale})} \mathop{\mathit{Sh}}\nolimits (Z_{\acute{e}tale}) \]

is an equivalence of categories.

Proof. We will work with the $2$-fibre product category as constructed in Categories, Example 4.31.3. The functor sends $\mathcal{F}$ to the triple $(f^{-1}\mathcal{F}, i^{-1}\mathcal{F}, c)$ where $c : j^{-1}f^{-1}\mathcal{F} \to g^{-1}i^{-1}\mathcal{F}$ is the canonical isomorphism. We will construct a quasi-inverse functor. Let $(\mathcal{F}', \mathcal{G}, \alpha )$ be an object of the right hand side of the arrow. We obtain an isomorphism

\[ i^{-1}f_*\mathcal{F}' = g_*j^{-1}\mathcal{F}' \xrightarrow {g_*\alpha } g_*g^{-1}\mathcal{G} \]

The first equality is Lemma 58.90.5. Using this we obtain maps $i_*\mathcal{G} \to i_*g_*g^{-1}\mathcal{G}$ and $f'_*\mathcal{F}' \to i_*g_*g^{-1}\mathcal{G}$. We set

\[ \mathcal{F} = f_*\mathcal{F}' \times _{i_*g_*g^{-1}\mathcal{G}} i_*\mathcal{G} \]

and we claim that $\mathcal{F}$ is an object of the left hand side of the arrow whose image in the right hand side is isomorphic to the triple we started out with. Let us compute the stalk of $\mathcal{F}$ at a geometric point $\overline{x}$ of $X$.

If $\overline{x}$ is not in $Z$, then on the one hand $\overline{x}$ comes from a unique geometric point $\overline{x}'$ of $X'$ and $\mathcal{F}'_{\overline{x}'} = (f_*\mathcal{F}')_{\overline{x}}$ and on the other hand we have $(i_*\mathcal{G})_{\overline{x}}$ and $(i_*g_*g^{-1}\mathcal{G})_{\overline{x}}$ are singletons. Hence we see that $\mathcal{F}_{\overline{x}}$ equals $\mathcal{F}'_{\overline{x}'}$.

If $\overline{x}$ is in $Z$, i.e., $\overline{x}$ is the image of a geometric point $\overline{z}$ of $Z$, then we obtain $(i_*\mathcal{G})_{\overline{x}} = \mathcal{G}_{\overline{z}}$ and

\[ (i_*g_*g^{-1}\mathcal{G})_{\overline{x}} = (g_*g^{-1}\mathcal{G})_{\overline{z}} = \Gamma (E_{\overline{z}}, g^{-1}\mathcal{G}|_{E_{\overline{z}}}) \]

(by the proper base change for pushforward used above) and similarly

\[ (f_*\mathcal{F}')_{\overline{x}} = \Gamma (X'_{\overline{x}}, \mathcal{F}'|_{X'_{\overline{x}}}) \]

Since we have the identification $E_{\overline{z}} = X'_{\overline{x}}$ and since $\alpha $ defines an isomorphism between the sheaves $\mathcal{F}'|_{X'_{\overline{x}}}$ and $g^{-1}\mathcal{G}|_{E_{\overline{z}}}$ we conclude that we get

\[ \mathcal{F}_{\overline{x}} = \mathcal{G}_{\overline{z}} \]

in this case.

To finish the proof, we observe that there are canonical maps $i^{-1}\mathcal{F} \to \mathcal{G}$ and $f^{-1}\mathcal{F} \to \mathcal{F}'$ compatible with $\alpha $ which on stalks produce the isomorphisms we saw above. We omit the careful construction of these maps. $\square$

Lemma 58.103.8. Let $S$ be a scheme. Then the category fibred in groupoids

\[ p : \mathcal{S} \longrightarrow (\mathit{Sch}/S)_ h \]

whose fibre category over $U$ is the category $\mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale})$ of sheaves on the small étale site of $U$ is a stack in groupoids.

Proof. To prove the lemma we will check conditions (1), (2), and (3) of More on Flatness, Lemma 38.37.13.

Condition (1) holds because we have glueing for sheaves (and Zariski coverings are étale coverings). See Sites, Lemma 7.26.4.

To see condition (2), suppose that $f : X \to Y$ is a surjective, flat, proper morphism of finite presentation over $S$ with $Y$ affine. Then we have descent for $\{ X \to Y\} $ by either Lemma 58.103.5 or Lemma 58.103.3.

Condition (3) follows immediately from the more general Lemma 58.103.7. $\square$


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