The Stacks project

58.96 Comparing ph and étale topologies

A model for this section is the section on the comparison of the usual topology and the qc topology on locally compact topological spaces as discussed in Cohomology on Sites, Section 21.30. We first review some material from Topologies, Sections 34.11 and 34.4.

Let $S$ be a scheme and let $(\mathit{Sch}/S)_{ph}$ be a ph site. On the same underlying category we have a second topology, namely the étale topology, and hence a second site $(\mathit{Sch}/S)_{\acute{e}tale}$. The identity functor $(\mathit{Sch}/S)_{\acute{e}tale}\to (\mathit{Sch}/S)_{ph}$ is continuous (by More on Morphisms, Lemma 37.44.7 and Topologies, Lemma 34.7.2) and defines a morphism of sites

\[ \epsilon _ S : (\mathit{Sch}/S)_{ph} \longrightarrow (\mathit{Sch}/S)_{\acute{e}tale} \]

See Cohomology on Sites, Section 21.26. Please note that $\epsilon _{S, *}$ is the identity functor on underlying presheaves and that $\epsilon _ S^{-1}$ associates to an étale sheaf the ph sheafification. Let $S_{\acute{e}tale}$ be the small étale site. There is a morphism of sites

\[ \pi _ S : (\mathit{Sch}/S)_{\acute{e}tale}\longrightarrow S_{\acute{e}tale} \]

given by the continuous functor $S_{\acute{e}tale}\to (\mathit{Sch}/S)_{\acute{e}tale}$, $U \mapsto U$. Namely, $S_{\acute{e}tale}$ has fibre products and a final object and the functor above commutes with these and Sites, Proposition 7.14.7 applies.

Lemma 58.96.1. With notation as above. Let $\mathcal{F}$ be a sheaf on $S_{\acute{e}tale}$. The rule

\[ (\mathit{Sch}/S)_{ph} \longrightarrow \textit{Sets},\quad (f : X \to S) \longmapsto \Gamma (X, f_{small}^{-1}\mathcal{F}) \]

is a sheaf and a fortiori a sheaf on $(\mathit{Sch}/S)_{\acute{e}tale}$. In fact this sheaf is equal to $\pi _ S^{-1}\mathcal{F}$ on $(\mathit{Sch}/S)_{\acute{e}tale}$ and $\epsilon _ S^{-1}\pi _ S^{-1}\mathcal{F}$ on $(\mathit{Sch}/S)_{ph}$.

Proof. The statement about the étale topology is the content of Lemma 58.39.2. To finish the proof it suffices to show that $\pi _ S^{-1}\mathcal{F}$ is a sheaf for the ph topology. By Topologies, Lemma 34.8.15 it suffices to show that given a proper surjective morphism $V \to U$ of schemes over $S$ we have an equalizer diagram

\[ \xymatrix{ (\pi _ S^{-1}\mathcal{F})(U) \ar[r] & (\pi _ S^{-1}\mathcal{F})(V) \ar@<1ex>[r] \ar@<-1ex>[r] & (\pi _ S^{-1}\mathcal{F})(V \times _ U V) } \]

Set $\mathcal{G} = \pi _ S^{-1}\mathcal{F}|_{U_{\acute{e}tale}}$. Consider the commutative diagram

\[ \xymatrix{ V \times _ U V \ar[r] \ar[rd]_ g \ar[d] & V \ar[d]^ f \\ V \ar[r]^ f & U } \]

We have

\[ (\pi _ S^{-1}\mathcal{F})(V) = \Gamma (V, f^{-1}\mathcal{G}) = \Gamma (U, f_*f^{-1}\mathcal{G}) \]

where we use $f_*$ and $f^{-1}$ to denote functorialities between small étale sites. Second, we have

\[ (\pi _ S^{-1}\mathcal{F})(V \times _ U V) = \Gamma (V \times _ U V, g^{-1}\mathcal{G}) = \Gamma (U, g_*g^{-1}\mathcal{G}) \]

The two maps in the equalizer diagram come from the two maps

\[ f_*f^{-1}\mathcal{G} \longrightarrow g_*g^{-1}\mathcal{G} \]

Thus it suffices to prove $\mathcal{G}$ is the equalizer of these two maps of sheaves. Let $\overline{u}$ be a geometric point of $U$. Set $\Omega = \mathcal{G}_{\overline{u}}$. Taking stalks at $\overline{u}$ by Lemma 58.87.4 we obtain the two maps

\[ H^0(V_{\overline{u}}, \underline{\Omega }) \longrightarrow H^0((V \times _ U V)_{\overline{u}}, \underline{\Omega }) = H^0(V_{\overline{u}} \times _{\overline{u}} V_{\overline{u}}, \underline{\Omega }) \]

where $\underline{\Omega }$ indicates the constant sheaf with value $\Omega $. Of course these maps are the pullback by the projection maps. Then it is clear that the sections coming from pullback by projection onto the first factor are constant on the fibres of the first projection, and sections coming from pullback by projection onto the first factor are constant on the fibres of the first projection. The sections in the intersection of the images of these pullback maps are constant on all of $V_{\overline{u}} \times _{\overline{u}} V_{\overline{u}}$, i.e., these come from elements of $\Omega $ as desired. $\square$

In the situation of Lemma 58.96.1 the composition of $\epsilon _ S$ and $\pi _ S$ and the equality determine a morphism of sites

\[ a_ S : (\mathit{Sch}/S)_{ph} \longrightarrow S_{\acute{e}tale} \]

Lemma 58.96.2. With notation as above. Let $f : X \to Y$ be a morphism of $(\mathit{Sch}/S)_{ph}$. Then there are commutative diagrams of topoi

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

and

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

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

Proof. The commutativity of the diagrams follows from the discussion in Topologies, Section 34.11. $\square$

Lemma 58.96.3. In Lemma 58.96.2 if $f$ is proper, then we have $a_ Y^{-1} \circ f_{small, *} = f_{big, ph, *} \circ a_ X^{-1}$.

Proof. You can prove this by repeating the proof of Lemma 58.93.5 part (1); we will instead deduce the result from this. As $\epsilon _{Y, *}$ is the identity functor on underlying presheaves, it reflects isomorphisms. The description in Lemma 58.96.1 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 58.93.5. $\square$

Lemma 58.96.4. Consider the comparison morphism $\epsilon : (\mathit{Sch}/S)_{ph} \to (\mathit{Sch}/S)_{\acute{e}tale}$. Let $\mathcal{P}$ denote the class of proper morphisms of schemes. For $X$ in $(\mathit{Sch}/S)_{\acute{e}tale}$ denote $\mathcal{A}'_ X \subset \textit{Ab}((\mathit{Sch}/X)_{\acute{e}tale})$ the full subcategory consisting of sheaves of the form $\pi _ X^{-1}\mathcal{F}$ where $\mathcal{F}$ is a torsion abelian sheaf on $X_{\acute{e}tale}$ Then Cohomology on Sites, Properties (1), (2), (3), (4), and (5) of Cohomology on Sites, Situation 21.29.1 hold.

Proof. We first show that $\mathcal{A}'_ X \subset \textit{Ab}((\mathit{Sch}/X)_{\acute{e}tale})$ is a weak Serre subcategory by checking conditions (1), (2), (3), and (4) of Homology, Lemma 12.10.3. Parts (1), (2), (3) are immediate as $\pi _ X^{-1}$ is exact and fully faithful for example by Lemma 58.93.4. If $0 \to \pi _ X^{-1}\mathcal{F} \to \mathcal{G} \to \pi _ X^{-1}\mathcal{F}' \to 0$ is a short exact sequence in $\textit{Ab}((\mathit{Sch}/X)_{\acute{e}tale})$ then $0 \to \mathcal{F} \to \pi _{X, *}\mathcal{G} \to \mathcal{F}' \to 0$ is exact by Lemma 58.93.4. In particular we see that $\pi _{X, *}\mathcal{G}$ is an abelian torsion sheaf on $X_{\acute{e}tale}$. Hence $\mathcal{G} = \pi _ X^{-1}\pi _{X, *}\mathcal{G}$ is in $\mathcal{A}'_ X$ which checks the final condition.

Cohomology on Sites, Property (1) holds by the existence of fibre products of schemes and the fact that the base change of a proper morphism of schemes is a proper morphism of schemes, see Morphisms, Lemma 29.41.5.

Cohomology on Sites, Property (2) follows from the commutative diagram (3) in Topologies, Lemma 34.4.16.

Cohomology on Sites, Property (3) is Lemma 58.96.1.

Cohomology on Sites, Property (4) holds by Lemma 58.93.5 part (2) and the fact that $R^ if_{small}\mathcal{F}$ is torsion if $\mathcal{F}$ is an abelian torsion sheaf on $X_{\acute{e}tale}$, see Lemma 58.76.2.

Cohomology on Sites, Property (5) follows from More on Morphisms, Lemma 37.44.1 combined with the fact that a finite morphism is proper and a surjective proper morphism defines a ph covering, see Topologies, Lemma 34.8.6. $\square$

Lemma 58.96.5. With notation as above.

  1. For $X \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{ph})$ and an abelian torsion sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $\epsilon _{X, *}a_ X^{-1}\mathcal{F} = \pi _ X^{-1}\mathcal{F}$ and $R^ i\epsilon _{X, *}(a_ X^{-1}\mathcal{F}) = 0$ for $i > 0$.

  2. For a proper morphism $f : X \to Y$ in $(\mathit{Sch}/S)_{ph}$ and abelian torsion sheaf $\mathcal{F}$ on $X$ we have $a_ Y^{-1}(R^ if_{small, *}\mathcal{F}) = R^ if_{big, ph, *}(a_ X^{-1}\mathcal{F})$ for all $i$.

  3. For a scheme $X$ and $K$ in $D^+(X_{\acute{e}tale})$ with torsion cohomology sheaves the map $\pi _ X^{-1}K \to R\epsilon _{X, *}(a_ X^{-1}K)$ is an isomorphism.

  4. For a proper morphism $f : X \to Y$ of schemes and $K$ in $D^+(X_{\acute{e}tale})$ with torsion cohomology sheaves we have $a_ Y^{-1}(Rf_{small, *}K) = Rf_{big, ph, *}(a_ X^{-1}K)$.

Proof. By Lemma 58.96.4 the lemmas in Cohomology on Sites, Section 21.29 all apply to our current setting. To translate the results observe that the category $\mathcal{A}_ X$ of Cohomology on Sites, Lemma 21.29.2 is the full subcategory of $\textit{Ab}((\mathit{Sch}/X)_{ph})$ consisting of sheaves of the form $a_ X^{-1}\mathcal{F}$ where $\mathcal{F}$ is an abelian torsion sheaf on $X_{\acute{e}tale}$.

Part (1) is equivalent to $(V_ n)$ for all $n$ which holds by Cohomology on Sites, Lemma 21.29.8.

Part (2) follows by applying $\epsilon _ Y^{-1}$ to the conclusion of Cohomology on Sites, Lemma 21.29.3.

Part (3) follows from Cohomology on Sites, Lemma 21.29.8 part (1) because $\pi _ X^{-1}K$ is in $D^+_{\mathcal{A}'_ X}((\mathit{Sch}/X)_{\acute{e}tale})$ and $a_ X^{-1} = \epsilon _ X^{-1} \circ a_ X^{-1}$.

Part (4) follows from Cohomology on Sites, Lemma 21.29.8 part (2) for the same reason. $\square$

Lemma 58.96.6. Let $X$ be a scheme. For $K \in D^+(X_{\acute{e}tale})$ with torsion cohomology sheaves the map

\[ K \longrightarrow Ra_{X, *}a_ X^{-1}K \]

is an isomorphism with $a_ X : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_{ph}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ as above.

Proof. We first reduce the statement to the case where $K$ is given by a single 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 58.96.1 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 58.96.5 we have $R^ i\epsilon _{X, *}(a_ X^{-1}\mathcal{F}) = 0$ for $i > 0$ and $\epsilon _{X, *}a_ X^{-1}\mathcal{F} = \pi _ X^{-1}\mathcal{F}$. By Lemma 58.93.4 we have $R^ j\pi _{X, *}(\pi _ X^{-1}\mathcal{F}) = 0$ for $j > 0$. This concludes the proof. $\square$

Lemma 58.96.7. For a scheme $X$ and $a_ X : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_{ph}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ 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 58.96.6 by Cohomology on Sites, Remark 21.14.4. $\square$


Comments (3)

Comment #4602 by Rex on

Minor typo: "On the same underlying category with have a second topology, namely the étale topology"

Is there any section on comparison between etale cohomology and singular cohomology? If not, are there any plans to add such a section?

Comment #4604 by on

Singular cohomology is outside the ambit of the Stacks project... at least for now.


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