The Stacks project

82.8 Comparing ph and étale topologies

This section is the analogue of Étale Cohomology, Section 58.96.

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. On the category $\textit{Spaces}/X$ we consider the ph and étale topologies. The identity functor $(\textit{Spaces}/X)_{\acute{e}tale}\to (\textit{Spaces}/X)_{ph}$ is continuous as every étale covering is a ph covering by Topologies on Spaces, Lemma 71.8.2. Hence it defines a morphism of sites

\[ \epsilon _ X : (\textit{Spaces}/X)_{ph} \longrightarrow (\textit{Spaces}/X)_{\acute{e}tale} \]

by an application of Sites, Proposition 7.14.7. Please note that $\epsilon _{X, *}$ is the identity functor on underlying presheaves and that $\epsilon _ X^{-1}$ associates to an étale sheaf the ph sheafification. Consider the morphism of sites

\[ \pi _ X : (\textit{Spaces}/X)_{\acute{e}tale}\longrightarrow X_{spaces, {\acute{e}tale}} \]

comparing big and small étale sites, see Section 82.5. The composition determines a morphism of sites

\[ a_ X = \pi _ X \circ \epsilon _ X : (\textit{Spaces}/X)_{ph} \longrightarrow X_{spaces, {\acute{e}tale}} \]

If $\mathcal{H}$ is an abelian sheaf on $(\textit{Spaces}/X)_{ph}$, then we will write $H^ n_{ph}(U, \mathcal{H})$ for the cohomology of $\mathcal{H}$ over an object $U$ of $(\textit{Spaces}/X)_{ph}$.

Lemma 82.8.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.

  1. For $\mathcal{F} \in \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ we have $\epsilon _{X, *}a_ X^{-1}\mathcal{F} = \pi _ X^{-1}\mathcal{F}$ and $a_{X, *}a_ X^{-1}\mathcal{F} = \mathcal{F}$.

  2. For $\mathcal{F} \in \textit{Ab}(X_{\acute{e}tale})$ torsion we have $R^ i\epsilon _{X, *}(a_ X^{-1}\mathcal{F}) = 0$ for $i > 0$.

Proof. We have $a_ X^{-1}\mathcal{F} = \epsilon _ X^{-1} \pi _ X^{-1}\mathcal{F}$. By Lemma 82.5.1 the étale sheaf $\pi _ X^{-1}\mathcal{F}$ is a sheaf for the ph topology and therefore is equal to $a_ X^{-1}\mathcal{F}$ (as pulling back by $\epsilon _ X$ is given by ph sheafification). Recall moreover that $\epsilon _{X, *}$ is the identity on underlying presheaves. Now part (1) is immediate from the explicit description of $\pi _ X^{-1}$ in Lemma 82.5.1.

We will prove part (2) by reducing it to the case of schemes – see part (1) of Étale Cohomology, Lemma 58.96.5. This will “clearly work” as every algebraic space is étale locally a scheme. The details are given below but we urge the reader to skip the proof.

For an abelian sheaf $\mathcal{H}$ on $(\textit{Spaces}/X)_{ph}$ the higher direct image $R^ p\epsilon _{X, *}\mathcal{H}$ is the sheaf associated to the presheaf $U \mapsto H^ p_{ph}(U, \mathcal{H})$ on $(\textit{Spaces}/X)_{\acute{e}tale}$. See Cohomology on Sites, Lemma 21.7.4. Since every object of $(\textit{Spaces}/X)_{\acute{e}tale}$ has a covering by schemes, it suffices to prove that given $U/X$ a scheme and $\xi \in H^ p_{ph}(U, a_ X^{-1}\mathcal{F})$ we can find an étale covering $\{ U_ i \to U\} $ such that $\xi $ restricts to zero on $U_ i$. We have

\begin{align*} H^ p_{ph}(U, a_ X^{-1}\mathcal{F}) & = H^ p((\textit{Spaces}/U)_{ph}, (a_ X^{-1}\mathcal{F})|_{\textit{Spaces}/U}) \\ & = H^ p((\mathit{Sch}/U)_{ph}, (a_ X^{-1}\mathcal{F})|_{\mathit{Sch}/U}) \end{align*}

where the second identification is Lemma 82.3.1 and the first is a general fact about restriction (Cohomology on Sites, Lemma 21.7.1). Looking at the first paragraph and the corresponding result in the case of schemes (Étale Cohomology, Lemma 58.96.1) we conclude that the sheaf $(a_ X^{-1}\mathcal{F})|_{\mathit{Sch}/U}$ matches the pullback by the “schemes version of $a_ U$”. Therefore we can find an étale covering $\{ U_ i \to U\} $ such that our class dies in $H^ p((\mathit{Sch}/U_ i)_{ph}, (a_ X^{-1}\mathcal{F})|_{\mathit{Sch}/U_ i})$ for each $i$, see Étale Cohomology, Lemma 58.96.5 (the precise statement one should use here is that $V_ n$ holds for all $n$ which is the statement of part (2) for the case of schemes). Transporting back (using the same formulas as above but now for $U_ i$) we conclude $\xi $ restricts to zero over $U_ i$ as desired. $\square$

The hard work done in the case of schemes now tells us that étale and ph cohomology agree for torsion abelian sheaves coming from the small étale site.

Lemma 82.8.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $K \in D^+(X_{\acute{e}tale})$ with torsion cohomology sheaves the maps

\[ \pi _ X^{-1}K \longrightarrow R\epsilon _{X, *}a_ X^{-1}K \quad \text{and}\quad K \longrightarrow Ra_{X, *}a_ X^{-1}K \]

are isomorphisms with $a_ X : \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{ph}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ as above.

Proof. We only prove the second statement; the first is easier and proved in exactly the same manner. There is a reduction to the case where $K$ is given by a single torsion abelian sheaf. Namely, represent $K$ by a bounded below complex $\mathcal{F}^\bullet $ of torsion abelian sheaves. This is possible by Cohomology on Sites, Lemma 21.19.8. By the case of a sheaf we see that $\mathcal{F}^ n = a_{X, *} a_ X^{-1} \mathcal{F}^ n$ and that the sheaves $R^ qa_{X, *}a_ X^{-1}\mathcal{F}^ n$ are zero for $q > 0$. By Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) applied to $a_ X^{-1}\mathcal{F}^\bullet $ and the functor $a_{X, *}$ we conclude. From now on assume $K = \mathcal{F}$ where $\mathcal{F}$ is a torsion abelian sheaf.

By Lemma 82.8.1 we have $a_{X, *}a_ X^{-1}\mathcal{F} = \mathcal{F}$. Thus it suffices to show that $R^ qa_{X, *}a_ X^{-1}\mathcal{F} = 0$ for $q > 0$. For this we can use $a_ X = \epsilon _ X \circ \pi _ X$ and the Leray spectral sequence (Cohomology on Sites, Lemma 21.14.7). By Lemma 82.8.1 we have $R^ i\epsilon _{X, *}(a_ X^{-1}\mathcal{F}) = 0$ for $i > 0$. We have $\epsilon _{X, *}a_ X^{-1}\mathcal{F} = \pi _ X^{-1}\mathcal{F}$ and by Lemma 82.5.5 we have $R^ j\pi _{X, *}(\pi _ X^{-1}\mathcal{F}) = 0$ for $j > 0$. This concludes the proof. $\square$

Lemma 82.8.3. Let $S$ be a scheme and let $X$ be an algebraic space over $S$. With $a_ X : \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{ph}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ as above:

  1. $H^ q(X_{\acute{e}tale}, \mathcal{F}) = H^ q_{ph}(X, a_ X^{-1}\mathcal{F})$ for a torsion abelian sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$,

  2. $H^ q(X_{\acute{e}tale}, K) = H^ q_{ph}(X, a_ X^{-1}K)$ for $K \in D^+(X_{\acute{e}tale})$ with torsion cohomology sheaves

Example: if $A$ is a torsion abelian group, then $H^ q_{\acute{e}tale}(X, \underline{A}) = H^ q_{ph}(X, \underline{A})$.

Proof. This follows from Lemma 82.8.2 by Cohomology on Sites, Remark 21.14.4. $\square$

Lemma 82.8.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Then there are commutative diagrams of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{ph}) \ar[rr]_{f_{big, ph}} \ar[d]_{\epsilon _ X} & & \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/Y)_{ph}) \ar[d]^{\epsilon _ Y} \\ \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{\acute{e}tale}) \ar[rr]^{f_{big, {\acute{e}tale}}} & & \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/Y)_{\acute{e}tale}) } \]

and

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{ph}) \ar[rr]_{f_{big, ph}} \ar[d]_{a_ X} & & \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/Y)_{ph}) \ar[d]^{a_ Y} \\ \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \ar[rr]^{f_{small}} & & \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) } \]

with $a_ X = \pi _ X \circ \epsilon _ X$ and $a_ Y = \pi _ X \circ \epsilon _ X$.

Proof. This follows immediately from working out the definitions of the morphisms involved, see Topologies on Spaces, Section 71.8 and Section 82.5. $\square$

Lemma 82.8.5. In Lemma 82.8.4 if $f$ is proper, then we have

  1. $a_ Y^{-1} \circ f_{small, *} = f_{big, ph, *} \circ a_ X^{-1}$, and

  2. $a_ Y^{-1}(Rf_{small, *}K) = Rf_{big, ph, *}(a_ X^{-1}K)$ for $K$ in $D^+(X_{\acute{e}tale})$ with torsion cohomology sheaves.

Proof. Proof of (1). You can prove this by repeating the proof of Lemma 82.5.6 part (1); we will instead deduce the result from this. As $\epsilon _{Y, *}$ is the identity functor on underlying presheaves, it reflects isomorphisms. Lemma 82.8.1 shows that $\epsilon _{Y, *} \circ a_ Y^{-1} = \pi _ Y^{-1}$ and similarly for $X$. To show that the canonical map $a_ Y^{-1}f_{small, *}\mathcal{F} \to f_{big, ph, *}a_ X^{-1}\mathcal{F}$ is an isomorphism, it suffices to show that

\begin{align*} \pi _ Y^{-1}f_{small, *}\mathcal{F} & = \epsilon _{Y, *}a_ Y^{-1}f_{small, *}\mathcal{F} \\ & \to \epsilon _{Y, *}f_{big, ph, *}a_ X^{-1}\mathcal{F} \\ & = f_{big, {\acute{e}tale}, *} \epsilon _{X, *}a_ X^{-1}\mathcal{F} \\ & = f_{big, {\acute{e}tale}, *}\pi _ X^{-1}\mathcal{F} \end{align*}

is an isomorphism. This is part (1) of Lemma 82.5.6.

To see (2) we use that

\begin{align*} R\epsilon _{Y, *}Rf_{big, ph, *}a_ X^{-1}K & = Rf_{big, {\acute{e}tale}, *}R\epsilon _{X, *}a_ X^{-1}K \\ & = Rf_{big, {\acute{e}tale}, *}\pi _ X^{-1}K \\ & = \pi _ Y^{-1}Rf_{small, *}K \\ & = R\epsilon _{Y, *} a_ Y^{-1}Rf_{small, *}K \end{align*}

The first equality by the commutative diagram in Lemma 82.8.4 and Cohomology on Sites, Lemma 21.19.2. Then second equality is Lemma 82.8.2. The third is Lemma 82.5.6 part (2). The fourth is Lemma 82.8.2 again. Thus the base change map $a_ Y^{-1}(Rf_{small, *}K) \to Rf_{big, ph, *}(a_ X^{-1}K)$ induces an isomorphism

\[ R\epsilon _{Y, *}a_ Y^{-1}Rf_{small, *}K \to R\epsilon _{Y, *}Rf_{big, ph, *}a_ X^{-1}K \]

The proof is finished by the following remark: consider a map $\alpha : a_ Y^{-1}L \to M$ with $L$ in $D^+(Y_{\acute{e}tale})$ having torsion cohomology sheaves and $M$ in $D^+((\textit{Spaces}/Y)_{ph})$. If $R\epsilon _{Y, *}\alpha $ is an isomorphism, then $\alpha $ is an isomorphism. Namely, we show by induction on $i$ that $H^ i(\alpha )$ is an isomorphism. This is true for all sufficiently small $i$. If it holds for $i \leq i_0$, then we see that $R^ j\epsilon _{Y, *}H^ i(M) = 0$ for $j > 0$ and $i \leq i_0$ by Lemma 82.8.1 because $H^ i(M) = a_ Y^{-1}H^ i(L)$ in this range. Hence $\epsilon _{Y, *}H^{i_0 + 1}(M) = H^{i_0 + 1}(R\epsilon _{Y, *}M)$ by a spectral sequence argument. Thus $\epsilon _{Y, *}H^{i_0 + 1}(M) = \pi _ Y^{-1}H^{i_0 + 1}(L) = \epsilon _{Y, *}a_ Y^{-1}H^{i_0 + 1}(L)$. This implies $H^{i_0 + 1}(\alpha )$ is an isomorphism (because $\epsilon _{Y, *}$ reflects isomorphisms as it is the identity on underlying presheaves) as desired. $\square$


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