This section is the analogue for algebraic spaces of Étale Cohomology, Section 59.104.
In order to conveniently express our results we need some notation. Let $S$ be a scheme. Let $\mathcal{U} = \{ f_ i : X_ i \to X\} $ be a family of morphisms of algebraic spaces over $S$ 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
$\mathcal{F}_ i$ is in $\mathop{\mathit{Sh}}\nolimits (X_{i, {\acute{e}tale}})$, and
$\varphi _{ij} : \text{pr}_{0, small}^{-1} \mathcal{F}_ i \longrightarrow \text{pr}_{1, small}^{-1} \mathcal{F}_ j$ 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
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
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
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
in $\mathop{\mathit{Sh}}\nolimits ((X \times _ Y X)_{\acute{e}tale})$ such that the cocycle condition holds:
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.
Lemma 81.4.1. Let $S$ be a scheme. Let $\{ f_ i : X_ i \to X\} $ be an étale covering of algebraic spaces. The functor is an equivalence of categories.
\[ \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt }\{ f_ i : X_ i \to X\} \]
Proof. In Properties of Spaces, Section 66.18 we have defined a site $X_{spaces, {\acute{e}tale}}$ whose objects are algebraic spaces étale over $X$ with étale coverings. Moreover, we have a identifications $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) = \mathop{\mathit{Sh}}\nolimits (X_{spaces, {\acute{e}tale}})$ compatible with morphisms of algebraic spaces, i.e., compatible with pushforward and pullback. Hence the statement of the lemma follows from the much more general discussion in Sites, Section 7.26. $\square$
Lemma 81.4.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\{ Y_ i \to Y\} _{i \in I}$ be an étale covering of algebraic spaces. If for each $i \in I$ the functor is an equivalence of categories and for each $i, j \in I$ the functor is an equivalence of categories, then is an equivalence of categories.
\[ \mathop{\mathit{Sh}}\nolimits (Y_{i, {\acute{e}tale}}) \longrightarrow \text{descent data for étale sheaves wrt }\{ X \times _ Y Y_ i \to Y_ i\} \]
\[ \mathop{\mathit{Sh}}\nolimits ((Y_ i \times _ Y Y_ j)_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt } \{ X \times _ Y Y_ i \times _ Y Y_ j \to Y_ i \times _ Y Y_ j\} \]
\[ \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt }\{ X \to Y\} \]
Proof. Formal consequence of Lemma 81.4.1 and the definitions. $\square$
Lemma 81.4.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is representable (by schemes) and $f$ has one of the following properties: surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation Then is an equivalence of categories.
\[ \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt }\{ X \to Y\} \]
Proof. Each of the properties of morphisms of algebraic spaces mentioned in the statement of the lemma is preserved by arbitrary base change, see the lists in Spaces, Section 65.4. Thus we can apply Lemma 81.4.2 to see that we can work étale locally on $Y$. In this way we reduce to the case where $Y$ is a scheme; some details omitted. In this case $X$ is also a scheme and the result follows from Étale Cohomology, Lemma 59.104.2, 59.104.3, or 59.104.5. $\square$
Lemma 81.4.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\pi : X' \to X$ be a morphism of algebraic spaces. Assume
$f \circ \pi $ is representable (by schemes),
$f \circ \pi $ has one of the following properties: surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation.
Then
is an equivalence of categories.
Proof. Formal consequence of Lemma 81.4.3 and Stacks, Lemma 8.3.7. $\square$
Lemma 81.4.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which has one of the following properties: surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation. Then the functor is an equivalence of categories.
\[ \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt }\{ X \to Y\} \]
Proof. Observe that the base change of a proper surjective morphism is proper and surjective, see Morphisms of Spaces, Lemmas 67.40.3 and 67.5.5. Hence by Lemma 81.4.2 we may work étale locally on $Y$. Hence we reduce to $Y$ being an affine scheme; some details omitted.
Assume $Y$ is affine. By Lemma 81.4.4 it suffices to find a morphism $X' \to X$ where $X'$ is a scheme such that $X' \to Y$ is surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation.
In case $X \to Y$ is integral and surjective, we can take $X = X'$ as an integral morphism is representable.
If $f$ is proper and surjective, then the algebraic space $X$ is quasi-compact and separated, see Morphisms of Spaces, Section 67.8 and Lemma 67.4.9. Choose a scheme $X'$ and a surjective finite morphism $X' \to X$, see Limits of Spaces, Proposition 70.16.1. Then $X' \to Y$ is surjective and proper.
Finally, if $X \to Y$ is surjective and flat and locally of finite presentation then we can take an affine étale covering $\{ U_ i \to X\} $ and set $X'$ equal to the disjoint $\coprod U_ i$. $\square$
Lemma 81.4.6. Let $S$ be a scheme. Let $\{ f_ i : X_ i \to X\} $ be an fppf covering of algebraic spaces over $S$. The functor is an equivalence of categories.
\[ \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt }\{ f_ i : X_ i \to X\} \]
Proof. We have Lemma 81.4.5 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 35.34.5. Details omitted. $\square$
Lemma 81.4.7. Let $S$ be a scheme. Let $f : Y' \to Y$ be a proper morphism of algebraic spaces over $S$. Let $i : Z \to Y$ be a closed immersion. Set $E = Z \times _ Y Y'$. Picture If $f$ is an isomorphism over $Y \setminus Z$, then the functor is an equivalence of categories.
\[ \xymatrix{ E \ar[d]_ g \ar[r]_ j & Y' \ar[d]^ f \\ Z \ar[r]^ i & Y } \]
\[ \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (Y'_{\acute{e}tale}) \times _{\mathop{\mathit{Sh}}\nolimits (E_{\acute{e}tale})} \mathop{\mathit{Sh}}\nolimits (Z_{\acute{e}tale}) \]
Proof. Observe that $X = Y' \coprod Z \to Y$ is a proper surjective morphism. Thus it suffice to construct an equivalence of categories
compatible with pullback functors from $Y$ because then we can use Lemma 81.4.5 to conclude. Thus let $(\mathcal{G}', \mathcal{G}, \alpha )$ be an object of $\mathop{\mathit{Sh}}\nolimits (Y'_{\acute{e}tale}) \times _{\mathop{\mathit{Sh}}\nolimits (E_{\acute{e}tale})} \mathop{\mathit{Sh}}\nolimits (Z_{\acute{e}tale})$ with notation as in Categories, Example 4.31.3. Then we can consider the sheaf $\mathcal{F}$ on $X$ defined by taking $\mathcal{G}'$ on the summand $Y'$ and $\mathcal{G}$ on the summand $Z$. We have
The isomorphisms of the two pullbacks of $\mathcal{F}$ to this algebraic space are obvious over the summands $E$, $E$, $Z$. The interesting part of the proof is to find an isomorphism $\text{pr}_{0, small}^{-1}\mathcal{G}' \to \text{pr}_{1, small}^{-1}\mathcal{G}'$ over $Y' \times _ Y Y'$ satisfying the cocycle condition. However, our assumption that $Y' \to Y$ is an isomorphism over $Y \setminus Z$ implies that
is a surjective proper morphism. (It is in fact a finite morphism as it is the disjoint union of two closed immersions.) Hence it suffices to construct an isomorphism of the pullbacks of $\text{pr}_{0, small}^{-1}\mathcal{G}'$and $\text{pr}_{1, small}^{-1}\mathcal{G}'$ by $h_{small}$ satisfying a certain cocycle condition. For the diagonal, it is clear how to do this. And for the pullback to $E \times _ Z E$ we use that both sheaves pull back to the pullback of $\mathcal{G}$ by the morphism $E \times _ Z E \to Z$. We omit the details. $\square$
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