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 21.31. Moreover, this section is almost word for word the same as the section comparing the ph and étale topologies. We first review some material from Topologies, Sections 34.11 and 34.4 and More on Flatness, Section 38.34.
Let $S$ be a scheme and let $(\mathit{Sch}/S)_ h$ be an h 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 $(\mathit{Sch}/S)_{\acute{e}tale}\to (\mathit{Sch}/S)_ h$ is continuous (by More on Flatness, Lemma 38.34.6 and Topologies, Lemma 34.7.2) and defines a morphism of sites
See Cohomology on Sites, Section 21.27. Please note that $\epsilon _{S, *}$ is the identity functor on underlying presheaves and that $\epsilon _ S^{-1}$ associates to an étale sheaf the h sheafification. Let $S_{\acute{e}tale}$ be the small étale site. There is a morphism of sites
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 59.103.1. With notation as above. Let $\mathcal{F}$ be a sheaf on $S_{\acute{e}tale}$. The rule 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 $(\mathit{Sch}/S)_ h$.
\[ (\mathit{Sch}/S)_ h \longrightarrow \textit{Sets},\quad (f : X \to S) \longmapsto \Gamma (X, f_{small}^{-1}\mathcal{F}) \]
Proof. The statement about the étale topology is the content of Lemma 59.39.2. To finish the proof it suffices to show that $\pi _ S^{-1}\mathcal{F}$ is a sheaf for the h topology. However, in Lemma 59.102.1 we have shown that $\pi _ S^{-1}\mathcal{F}$ is a sheaf even in the stronger ph topology. $\square$
In the situation of Lemma 59.103.1 the composition of $\epsilon _ S$ and $\pi _ S$ and the equality determine a morphism of sites
Lemma 59.103.2. With notation as above. Let $f : X \to Y$ be a morphism of $(\mathit{Sch}/S)_ h$. Then there are commutative diagrams of topoi and with $a_ X = \pi _ X \circ \epsilon _ X$ and $a_ Y = \pi _ X \circ \epsilon _ X$.
\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_ h) \ar[rr]_{f_{big, h}} \ar[d]_{\epsilon _ X} & & \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/Y)_ h) \ar[d]^{\epsilon _ Y} \\ \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_{\acute{e}tale}) \ar[rr]^{f_{big, {\acute{e}tale}}} & & \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/Y)_{\acute{e}tale}) } \]
\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_ h) \ar[rr]_{f_{big, h}} \ar[d]_{a_ X} & & \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/Y)_ h) \ar[d]^{a_ Y} \\ \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \ar[rr]^{f_{small}} & & \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) } \]
Proof. The commutativity of the diagrams follows similarly to what was said in Topologies, Section 34.11. $\square$
Lemma 59.103.3. In Lemma 59.103.2 if $f$ is proper, then we have $a_ Y^{-1} \circ f_{small, *} = f_{big, h, *} \circ a_ X^{-1}$.
Proof. You can prove this by repeating the proof of Lemma 59.99.5 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 59.103.1 shows that $\epsilon _{Y, *} \circ a_ Y^{-1} = \pi _ Y^{-1}$ and similarly for $X$. To show that the canonical map $a_ Y^{-1}f_{small, *}\mathcal{F} \to f_{big, h, *}a_ X^{-1}\mathcal{F}$ is an isomorphism, it suffices to show that
is an isomorphism. This is part (1) of Lemma 59.99.5. $\square$
Lemma 59.103.4. Consider the comparison morphism $\epsilon : (\mathit{Sch}/S)_ h \to (\mathit{Sch}/S)_{\acute{e}tale}$. Let $\mathcal{P}$ denote the class of proper morphisms. 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 21.30.1 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 59.99.4. 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 59.99.4. 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, the fact that the base change of a proper morphism of schemes is a proper morphism of schemes, see Morphisms, Lemma 29.41.5, and the fact that the base change of a morphism of finite presentation is a morphism of finite presentation, see Morphisms, Lemma 29.21.4.
Cohomology on Sites, Property (2) follows from the commutative diagram (3) in Topologies, Lemma 34.4.17.
Cohomology on Sites, Property (3) is Lemma 59.103.1.
Cohomology on Sites, Property (4) holds by Lemma 59.99.5 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 59.78.2.
Cohomology on Sites, Property (5) is implied by More on Morphisms, Lemma 37.48.1 combined with the fact that a surjective finite locally free morphism is surjective, proper, and of finite presentation and hence defines a h-covering by More on Flatness, Lemma 38.34.7. $\square$
Lemma 59.103.5. With notation as above.
For $X \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{h})$ 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 $i > 0$.
For a proper morphism $f : X \to Y$ in $(\mathit{Sch}/S)_ h$ and abelian torsion sheaf $\mathcal{F}$ on $X$ we have $a_ Y^{-1}(R^ if_{small, *}\mathcal{F}) = R^ if_{big, h, *}(a_ X^{-1}\mathcal{F})$ for all $i$.
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.
For a proper morphism $f : X \to Y$ of schemes and $K$ in $D^+(X_{\acute{e}tale})$ with torsion cohomology sheaves we have $a_ Y^{-1}(Rf_{small, *}K) = Rf_{big, h, *}(a_ X^{-1}K)$.
Proof. By Lemma 59.103.4 the lemmas in Cohomology on Sites, Section 21.30 all apply to our current setting. To translate the results observe that the category $\mathcal{A}_ X$ of Cohomology on Sites, Lemma 21.30.2 is the full subcategory of $\textit{Ab}((\mathit{Sch}/X)_ h)$ 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 21.30.8.
Part (2) follows by applying $\epsilon _ Y^{-1}$ to the conclusion of Cohomology on Sites, Lemma 21.30.3.
Part (3) follows from Cohomology on Sites, Lemma 21.30.8 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 21.30.8 part (2) for the same reason. $\square$
Lemma 59.103.6. Let $X$ be a scheme. For $K \in D^+(X_{\acute{e}tale})$ with torsion cohomology sheaves the map is an isomorphism with $a_ X : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_ h) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ as above.
\[ K \longrightarrow Ra_{X, *}a_ X^{-1}K \]
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 59.103.1 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 59.103.5 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 59.99.4 we have $R^ j\pi _{X, *}(\pi _ X^{-1}\mathcal{F}) = 0$ for $j > 0$. This concludes the proof. $\square$
Lemma 59.103.7. For a scheme $X$ and $a_ X : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_ h) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ as above:
$H^ q(X_{\acute{e}tale}, \mathcal{F}) = H^ q_ h(X, a_ X^{-1}\mathcal{F})$ for a torsion abelian sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$,
$H^ q(X_{\acute{e}tale}, K) = H^ q_ h(X, a_ X^{-1}K)$ for $K \in D^+(X_{\acute{e}tale})$ with torsion cohomology sheaves.
Example: if $A$ is a torsion abelian group, then $H^ q_{\acute{e}tale}(X, \underline{A}) = H^ q_ h(X, \underline{A})$.
Proof. This follows from Lemma 59.103.6 by Cohomology on Sites, Remark 21.14.4. $\square$
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