The Stacks project

82.3 Transporting results from schemes

In this section we explain briefly how results for schemes imply results for (representable) algebraic spaces and (representable) morphisms of algebraic spaces. For quasi-coherent modules more is true (because étale cohomology of a quasi-coherent module over a scheme agrees with Zariski cohomology) and this has already been discussed in Cohomology of Spaces, Section 67.3.

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Now suppose that $X$ is representable by the scheme $X_0$ (awkward but temporary notation; we usually just say “$X$ is a scheme”). In this case $X$ and $X_0$ have the same small étale sites:

\[ X_{\acute{e}tale}= (X_0)_{\acute{e}tale} \]

This is pointed out in Properties of Spaces, Section 64.18. Moreover, if $f : X \to Y$ is a morphism of representable algebraic spaces over $S$ and if $f_0 : X_0 \to Y_0$ is a morphism of schemes representing $f$, then the induced morphisms of small étale topoi agree:

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \ar[rr]_{f_{small}} \ar@{=}[d] & & \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \ar@{=}[d] \\ \mathop{\mathit{Sh}}\nolimits ((X_0)_{\acute{e}tale}) \ar[rr]^{(f_0)_{small}} & & \mathop{\mathit{Sh}}\nolimits ((Y_0)_{\acute{e}tale}) } \]

See Properties of Spaces, Lemma 64.18.7 and Topologies, Lemma 34.4.16.

Thus there is absolutely no difference between étale cohomology of a scheme and the étale cohomology of the corresponding algebraic space. Similarly for higher direct images along morphisms of schemes. In fact, if $f : X \to Y$ is a morphism of algebraic spaces over $S$ which is representable (by schemes), then the higher direct images $R^ if_*\mathcal{F}$ of a sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ can be computed étale locally on $Y$ (Cohomology on Sites, Lemma 21.7.4) hence this often reduces computations and proofs to the case where $Y$ and $X$ are schemes.

We will use the above without further mention in this chapter. For other topologies the same thing is true; we state it explicitly as a lemma for cohomology here.

Lemma 82.3.1. Let $S$ be a scheme. Let $\tau \in \{ {\acute{e}tale}, fppf, ph\} $ (add more here). The inclusion functor

\[ (\mathit{Sch}/S)_\tau \longrightarrow (\textit{Spaces}/S)_\tau \]

is a special cocontinuous functor (Sites, Definition 7.29.2) and hence identifies topoi.

Proof. The conditions of Sites, Lemma 7.29.1 are immediately verified as our functor is fully faithful and as every algebraic space has an étale covering by schemes. $\square$


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