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 \input{preamble} % OK, start here. % \begin{document} \title{More on Cohomology of Spaces} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent In this chapter continues the discussion started in Cohomology of Spaces, Section \ref{spaces-cohomology-section-introduction}. One can also view this chapter as the analogue for algebraic spaces of the chapter on \'etale cohomology for schemes, see \'Etale Cohomology, Section \ref{etale-cohomology-section-introduction}. \medskip\noindent In fact, we intend this chapter to be mainly a translation of the results already proved for schemes into the language of algebraic spaces. Some of our results can be found in \cite{Kn}. \section{Conventions} \label{section-conventions} \noindent The standing assumption is that all schemes are contained in a big fppf site $\Sch_{fppf}$. And all rings $A$ considered have the property that $\Spec(A)$ is (isomorphic) to an object of this big site. \medskip\noindent Let $S$ be a scheme and let $X$ be an algebraic space over $S$. In this chapter and the following we will write $X \times_S X$ for the product of $X$ with itself (in the category of algebraic spaces over $S$), instead of $X \times X$. \section{Transporting results from schemes} \label{section-api} \noindent 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 \'etale cohomology of a quasi-coherent module over a scheme agrees with Zariski cohomology) and this has already been discussed in Cohomology of Spaces, Section \ref{spaces-cohomology-section-higher-direct-image}. \medskip\noindent 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 \'etale sites: $$X_\etale = (X_0)_\etale$$ This is pointed out in Properties of Spaces, Section \ref{spaces-properties-section-etale-site}. 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 \'etale topoi agree: $$\xymatrix{ \Sh(X_\etale) \ar[rr]_{f_{small}} \ar@{=}[d] & & \Sh(Y_\etale) \ar@{=}[d] \\ \Sh((X_0)_\etale) \ar[rr]^{(f_0)_{small}} & & \Sh((Y_0)_\etale) }$$ See Properties of Spaces, Lemma \ref{spaces-properties-lemma-functoriality-etale-site} and Topologies, Lemma \ref{topologies-lemma-morphism-big-small-etale}. \medskip\noindent Thus there is absolutely no difference between \'etale cohomology of a scheme and the \'etale 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 $X_\etale$ can be computed \'etale locally on $Y$ (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-higher-direct-images}) hence this often reduces computations and proofs to the case where $Y$ and $X$ are schemes. \medskip\noindent 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. \begin{lemma} \label{lemma-compare-cohomology-other-topologies} Let $S$ be a scheme. Let $\tau \in \{\etale, fppf, ph\}$ (add more here). The inclusion functor $$(\Sch/S)_\tau \longrightarrow (\textit{Spaces}/S)_\tau$$ is a special cocontinuous functor (Sites, Definition \ref{sites-definition-special-cocontinuous-functor}) and hence identifies topoi. \end{lemma} \begin{proof} The conditions of Sites, Lemma \ref{sites-lemma-equivalence} are immediately verified as our functor is fully faithful and as every algebraic space has an \'etale covering by schemes. \end{proof} \section{Proper base change} \label{section-proper-base-change} \noindent The proper base change theorem for algebraic spaces follows from the proper base change theorem for schemes and Chow's lemma with a little bit of work. \begin{lemma} \label{lemma-surjective-proper} Let $S$ be a scheme. Let $f : Y \to X$ be a surjective proper morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a sheaf on $X_\etale$. Then $\mathcal{F} \to f_*f^{-1}\mathcal{F}$ is injective with image the equalizer of the two maps $f_*f^{-1}\mathcal{F} \to g_*g^{-1}\mathcal{F}$ where $g$ is the structure morphism $g : Y \times_X Y \to X$. \end{lemma} \begin{proof} For any surjective morphism $f : Y \to X$ of algebraic spaces over $S$, the map $\mathcal{F} \to f_*f^{-1}\mathcal{F}$ is injective. Namely, if $\overline{x}$ is a geometric point of $X$, then we choose a geometric point $\overline{y}$ of $Y$ lying over $\overline{x}$ and we consider $$\mathcal{F}_{\overline{x}} \to (f_*f^{-1}\mathcal{F})_{\overline{x}} \to (f^{-1}\mathcal{F})_{\overline{y}} = \mathcal{F}_{\overline{x}}$$ See Properties of Spaces, Lemma \ref{spaces-properties-lemma-stalk-pullback} for the last equality. \medskip\noindent The second statement is local on $X$ in the \'etale topology, hence we may and do assume $Y$ is an affine scheme. \medskip\noindent Choose a surjective proper morphism $Z \to Y$ where $Z$ is a scheme, see Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-weak-chow}. The result for $Z \to X$ implies the result for $Y \to X$. Since $Z \to X$ is a surjective proper morphism of schemes and hence a ph covering (Topologies, Lemma \ref{topologies-lemma-surjective-proper-ph}) the result for $Z \to X$ follows from \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-describe-pullback-pi-ph} (in fact it is in some sense equivalent to this lemma). \end{proof} \begin{lemma} \label{lemma-h0-proper-over-henselian-pair} Let $(A, I)$ be a henselian pair. Let $X$ be an algebraic space over $A$ such that the structure morphism $f : X \to \Spec(A)$ is proper. Let $i : X_0 \to X$ be the inclusion of $X \times_{\Spec(A)} \Spec(A/I)$. For any sheaf $\mathcal{F}$ on $X_\etale$ we have $\Gamma(X, \mathcal{F}) = \Gamma(Z, i^{-1}\mathcal{F})$. \end{lemma} \begin{proof} Choose a surjective proper morphism $Y \to X$ where $Y$ is a scheme, see Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-weak-chow}. Consider the diagram $$\xymatrix{ \Gamma(X_0, \mathcal{F}_0) \ar[r] \ar[d] & \Gamma(Y_0, \mathcal{G}_0) \ar@<1ex>[r] \ar@<-1ex>[r] \ar[d] & \Gamma((Y \times_X Y)_0, \mathcal{H}_0) \ar[d] \\ \Gamma(X, \mathcal{F}) \ar[r] & \Gamma(Y, \mathcal{G}) \ar@<1ex>[r] \ar@<-1ex>[r] & \Gamma(Y \times_X Y, \mathcal{H}) }$$ Here $\mathcal{G}$, resp.\ $\mathcal{H}$ is the pullbackf or $\mathcal{F}$ to $Y$, resp.\ $Y \times_X Y$ and the index $0$ indicates base change to $\Spec(A/I)$. By the case of schemes (\'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-h0-proper-over-henselian-pair}) we see that the middle and right vertical arrows are bijective. By Lemma \ref{lemma-surjective-proper} it follows that the left one is too. \end{proof} \begin{lemma} \label{lemma-h0-proper-over-henselian-local} Let $A$ be a henselian local ring. Let $X$ be an algebraic space over $A$ such that $f : X \to \Spec(A)$ be a proper morphism. Let $X_0 \subset X$ be the fibre of $f$ over the closed point. For any sheaf $\mathcal{F}$ on $X_\etale$ we have $\Gamma(X, \mathcal{F}) = \Gamma(X_0, \mathcal{F}|_{X_0})$. \end{lemma} \begin{proof} This is a special case of Lemma \ref{lemma-h0-proper-over-henselian-pair}. \end{proof} \begin{lemma} \label{lemma-proper-base-change-f-star} Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y' \to Y$ be a morphisms of algebraic spaces over $S$. Assume $f$ is proper. Set $X' = Y' \times_Y X$ with projections $f' : X' \to Y'$ and $g' : X' \to X$. Let $\mathcal{F}$ be any sheaf on $X_\etale$. Then $g^{-1}f_*\mathcal{F} = f'_*(g')^{-1}\mathcal{F}$. \end{lemma} \begin{proof} The question is \'etale local on $Y'$. Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Choose a scheme $V'$ and a surjective \'etale morphism $V' \to V \times_Y Y'$. Then we may replace $Y'$ by $V'$ and $Y$ by $V$. Hence we may assume $Y$ and $Y'$ are schemes. Then we may work Zariski locally on $Y$ and $Y'$ and hence we may assume $Y$ and $Y'$ are affine schemes. \medskip\noindent Assume $Y$ and $Y'$ are affine schemes. Choose a surjective proper morphism $h_1 : X_1 \to X$ where $X_1$ is a scheme, see Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-weak-chow}. Set $X_2 = X_1 \times_X X_1$ and denote $h_2 : X_2 \to X$ the structure morphism. Observe this is a scheme. By the case of schemes (\'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-proper-base-change-f-star}) we know the lemma is true for the cartesian diagrams $$\vcenter{ \xymatrix{ X'_1 \ar[r] \ar[d] & X_1 \ar[d] \\ Y' \ar[r] & Y } } \quad\text{and}\quad \vcenter{ \xymatrix{ X'_2 \ar[r] \ar[d] & X_2 \ar[d] \\ Y' \ar[r] & Y } }$$ and the sheaves $\mathcal{F}_i = (X_i \to X)^{-1}\mathcal{F}$. By Lemma \ref{lemma-surjective-proper} we have an exact sequence $0 \to \mathcal{F} \to h_{1, *}\mathcal{F}_1 \to h_{2, *}\mathcal{F}_2$ and similarly for $(g')^{-1}\mathcal{F}$ because $X'_2 = X'_1 \times_{X'} X'_1$. Hence we conlude that the lemma is true (some details omitted). \end{proof} \noindent Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Let $\overline{x} : \Spec(k) \to S$ be a geometric point. The fibre of $f$ at $\overline{x}$ is the algebraic space $Y_{\overline{x}} = \Spec(k) \times_{\overline{x}, X} Y$ over $\Spec(k)$. If $\mathcal{F}$ is a sheaf on $Y_\etale$, then denote $\mathcal{F}_{\overline{x}} = p^{-1}\mathcal{F}$ the pullback of $\mathcal{F}$ to $(Y_{\overline{x}})_\etale$. Here $p : Y_{\overline{x}} \to Y$ is the projection. In the following we will consider the set $\Gamma(Y_{\overline{x}}, \mathcal{F}_{\overline{x}})$. \begin{lemma} \label{lemma-proper-pushforward-stalk} Let $S$ be a scheme. Let $f : Y \to X$ be a proper morphism of algebraic spaces over $S$. Let $\overline{x} \to X$ be a geometric point. For any sheaf $\mathcal{F}$ on $Y_\etale$ the canonical map $$(f_*\mathcal{F})_{\overline{x}} \longrightarrow \Gamma(Y_{\overline{x}}, \mathcal{F}_{\overline{x}})$$ is bijective. \end{lemma} \begin{proof} This is a special case of Lemma \ref{lemma-proper-base-change-f-star}. \end{proof} \begin{theorem} \label{theorem-proper-base-change} Let $S$ be a scheme. Let $$\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }$$ be a cartesian square of algebraic spaces over $S$. Assume $f$ is proper. Let $\mathcal{F}$ be an abelian torsion sheaf on $X_\etale$. Then the base change map $$g^{-1}Rf_*\mathcal{F} \longrightarrow Rf'_*(g')^{-1}\mathcal{F}$$ is an isomorphism. \end{theorem} \begin{proof} This proof repeats a few of the arguments given in the proof of the proper base change theorem for schemes. See \'Etale Cohomology, Section \ref{etale-cohomology-section-proper-base-change} for more details. \medskip\noindent The statement is \'etale local on $Y'$ and $Y$, hence we may assume both $Y$ and $Y'$ are affine schemes. Observe that this in particular proves the theorem in case $f$ is representable (we will use this below). \medskip\noindent For every $n \geq 1$ let $\mathcal{F}[n]$ be the subsheaf of sections of $\mathcal{F}$ annihilated by $n$. Then $\mathcal{F} = \colim \mathcal{F}[n]$. By Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-colimit-cohomology} the functors $g^{-1}R^pf_*$ and $R^pf'_*(g')^{-1}$ commute with filtered colimits. Hence it suffices to prove the theorem if $\mathcal{F}$ is killed by $n$. \medskip\noindent Let $\mathcal{F} \to \mathcal{I}^\bullet$ be a resolution by injective sheaves of $\mathbf{Z}/n\mathbf{Z}$-modules. Observe that $g^{-1}f_*\mathcal{I}^\bullet = f'_*(g')^{-1}\mathcal{I}^\bullet$ by Lemma \ref{lemma-proper-base-change-f-star}. Applying Leray's acyclicity lemma (Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity}) we conclude it suffices to prove $R^pf'_*(g')^{-1}\mathcal{I}^m = 0$ for $p > 0$ and $m \in \mathbf{Z}$. \medskip\noindent Choose a surjective proper morphism $h : Z \to X$ where $Z$ is a scheme, see Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-weak-chow}. Choose an injective map $h^{-1}\mathcal{I}^m \to \mathcal{J}$ where $\mathcal{J}$ is an injective sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules on $Z_\etale$. Since $h$ is surjective the map $\mathcal{I}^m \to h_*\mathcal{J}$ is injective (see Lemma \ref{lemma-surjective-proper}). Since $\mathcal{I}^m$ is injective we see that $\mathcal{I}^m$ is a direct summand of $h_*\mathcal{J}$. Thus it suffices to prove the desired vanishing for $h_*\mathcal{J}$. \medskip\noindent Denote $h'$ the base change by $g$ and denote $g'' : Z' \to Z$ the projection. There is a spectral sequence $$E_2^{p, q} = R^pf'_* R^qh'_* (g'')^{-1}\mathcal{J}$$ converging to $R^{p + q}(f' \circ h')_*(g'')^{-1}\mathcal{J}$. Since $h$ and $f \circ h$ are representable (by schemes) we know the result we want holds for them. Thus in the spectral sequence we see that $E_2^{p, q} = 0$ for $q > 0$ and $R^{p + q}(f' \circ h')_*(g'')^{-1}\mathcal{J} = 0$ for $p + q > 0$. It follows that $E_2^{p, 0} = 0$ for $p > 0$. Now $$E_2^{p, 0} = R^pf'_* h'_* (g'')^{-1}\mathcal{J} = R^pf'_* (g')^{-1}h_*\mathcal{J}$$ by Lemma \ref{lemma-proper-base-change-f-star}. This finishes the proof. \end{proof} \begin{lemma} \label{lemma-proper-base-change} Let $S$ be a scheme. Let $$\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }$$ be a cartesian square of algebraic spaces over $S$. Assume $f$ is proper. Let $E \in D^+(X_\etale)$ have torsion cohomology sheaves. Then the base change map $g^{-1}Rf_*E \to Rf'_*(g')^{-1}E$ is an isomorphism. \end{lemma} \begin{proof} This is a simple consequence of the proper base change theorem (Theorem \ref{theorem-proper-base-change}) using the spectral sequences $$E_2^{p, q} = R^pf_*H^q(E) \quad\text{and}\quad {E'}_2^{p, q} = R^pf'_*(g')^{-1}H^q(E)$$ converging to $R^nf_*E$ and $R^nf'_*(g')^{-1}E$. The spectral sequences are constructed in Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}. Some details omitted. \end{proof} \begin{lemma} \label{lemma-proper-base-change-stalk} Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of algebraic spaces. Let $\overline{y} \to Y$ be a geometric point. \begin{enumerate} \item For a torsion abelian sheaf $\mathcal{F}$ on $X_\etale$ we have $(R^nf_*\mathcal{F})_{\overline{y}} = H^n_\etale(X_{\overline{y}}, \mathcal{F}_{\overline{y}})$. \item For $E \in D^+(X_\etale)$ with torsion cohomology sheaves we have $(R^nf_*E)_{\overline{y}} = H^n_\etale(X_{\overline{y}}, E_{\overline{y}})$. \end{enumerate} \end{lemma} \begin{proof} In the statement, $\mathcal{F}_{\overline{y}}$ denotes the pullback of $\mathcal{F}$ to $X_{\overline{y}} = \overline{y} \times_Y X$. Since pulling back by $\overline{y} \to Y$ produces the stalk of $\mathcal{F}$, the first statement of the lemma is a special case of Theorem \ref{theorem-proper-base-change}. The second one is a special case of Lemma \ref{lemma-proper-base-change}. \end{proof} \begin{lemma} \label{lemma-base-change-separably-closed} Let $k \subset k'$ be an extension of separably closed fields. Let $X$ be a proper algebraic space over $k$. Let $\mathcal{F}$ be a torsion abelian sheaf on $X$. Then the map $H^q_\etale(X, \mathcal{F}) \to H^q_\etale(X_{k'}, \mathcal{F}|_{X_{k'}})$ is an isomorphism for $q \geq 0$. \end{lemma} \begin{proof} This is a special case of Theorem \ref{theorem-proper-base-change}. \end{proof} \section{Comparing big and small topoi} \label{section-compare} \noindent Let $S$ be a scheme and let $X$ be an algebraic space over $S$. In Topologies on Spaces, Lemma \ref{spaces-topologies-lemma-at-the-bottom-etale} we have introduced comparison morphisms $\pi_X : (\textit{Spaces}/X)_\etale \to X_{spaces, \etale}$ and $i_X : \Sh(X_\etale) \to \Sh((\textit{Spaces}/X)_\etale)$ with $\pi_X \circ i_X = \text{id}$ as morphisms of topoi and $\pi_{X, *} = i_X^{-1}$. More generally, if $f : Y \to X$ is an object of $(\textit{Spaces}/X)_\etale$, then there is a morphism $i_f : \Sh(Y_\etale) \to \Sh((\textit{Spaces}/X)_\etale)$ such that $f_{small} = \pi_X \circ i_f$, see Topologies on Spaces, Lemmas \ref{spaces-topologies-lemma-put-in-T-etale} and \ref{spaces-topologies-lemma-morphism-big-small-etale}. In Topologies on Spaces, Remark \ref{spaces-topologies-remark-change-topologies-ringed} we have extended these to a morphism of ringed sites $$\pi_X : ((\textit{Spaces}/X)_\etale, \mathcal{O}) \to (X_{spaces, \etale}, \mathcal{O}_X)$$ and morphisms of ringed topoi $$i_X : (\Sh(X_\etale), \mathcal{O}_X) \to (\Sh((\textit{Spaces}/X)_\etale), \mathcal{O})$$ and $$i_f : (\Sh(Y_\etale), \mathcal{O}_Y) \to (\Sh((\textit{Spaces}/X)_\etale, \mathcal{O}))$$ Note that the restriction $i_X^{-1} = \pi_{X, *}$ (see Topologies, Definition \ref{topologies-definition-restriction-small-etale}) transforms $\mathcal{O}$ into $\mathcal{O}_X$. Similarly, $i_f^{-1}$ transforms $\mathcal{O}$ into $\mathcal{O}_Y$. See Topologies on Spaces, Remark \ref{spaces-topologies-remark-change-topologies-ringed}. Hence $i_X^*\mathcal{F} = i_X^{-1}\mathcal{F}$ and $i_f^*\mathcal{F} = i_f^{-1}\mathcal{F}$ for any $\mathcal{O}$-module $\mathcal{F}$ on $(\textit{Spaces}/X)_\etale$. In particular $i_X^*$ and $i_f^*$ are exact functors. The functor $i_X^*$ is often denoted $\mathcal{F} \mapsto \mathcal{F}|_{X_\etale}$ (and this does not conflict with the notation in Topologies on Spaces, Definition \ref{spaces-topologies-definition-restriction-small-etale}). \begin{lemma} \label{lemma-describe-pullback} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a sheaf on $X_\etale$. Then $\pi_X^{-1}\mathcal{F}$ is given by the rule $$(\pi_X^{-1}\mathcal{F})(Y) = \Gamma(Y_\etale, f_{small}^{-1}\mathcal{F})$$ for $f : Y \to X$ in $(\textit{Spaces}/X)_\etale$. Moreover, $\pi_Y^{-1}\mathcal{F}$ satisfies the sheaf condition with respect to smooth, syntomic, fppf, fpqc, and ph coverings. \end{lemma} \begin{proof} Since pullback is transitive and $f_{small} = \pi_X \circ i_f$ (see above) we see that $i_f^{-1} \pi_X^{-1}\mathcal{F} = f_{small}^{-1}\mathcal{F}$. This shows that $\pi_X^{-1}$ has the description given in the lemma. \medskip\noindent To prove that $\pi_X^{-1}\mathcal{F}$ is a sheaf for the ph topology it suffices by Topologies on Spaces, Lemma \ref{spaces-topologies-lemma-characterize-sheaf} to show that for a surjective proper morphism $V \to U$ of algebraic spaces over $X$ we have $(\pi_X^{-1}\mathcal{F})(U)$ is the equalizer of the two maps $(\pi_X^{-1}\mathcal{F})(V) \to (\pi_X^{-1}\mathcal{F})(V \times_U V)$. This we have seen in Lemma \ref{lemma-surjective-proper}. \medskip\noindent The case of smooth, syntomic, fppf coverings follows from the case of ph coverings by Topologies on Spaces, Lemma \ref{spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf-ph}. \medskip\noindent Let $\mathcal{U} = \{U_i \to U\}_{i \in I}$ be an fpqc covering of algebraic spaces over $X$. Let $s_i \in (\pi_X^{-1}\mathcal{F})(U_i)$ be sections which agree over $U_i \times_U U_j$. We have to prove there exists a unique $s \in (\pi_X^{-1}\mathcal{F})(U)$ restricting to $s_i$ over $U_i$. Case I: $U$ and $U_i$ are schemes. This case follows from \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-describe-pullback}. Case II: $U$ is a scheme. Here we choose surjective \'etale morphisms $T_i \to U_i$ where $T_i$ is a scheme. Then $\mathcal{T} = \{T_i \to U\}$ is an fpqc covering by schemes and by case I the result holds for $\mathcal{T}$. We omit the verification that this implies the result for $\mathcal{U}$. Case III: general case. Let $W \to U$ be a surjective \'etale morphism, where $W$ is a scheme. Then $\mathcal{W} = \{U_i \times_U W \to W\}$ is an fpqc covering (by algebraic spaces) of the scheme $W$. By case II the result hold for $\mathcal{W}$. We omit the verification that this implies the result for $\mathcal{U}$. \end{proof} \begin{lemma} \label{lemma-compare-injectives} Let $S$ be a scheme. Let $Y \to X$ be a morphism of $(\textit{Spaces}/S)_\etale$. \begin{enumerate} \item If $\mathcal{I}$ is injective in $\textit{Ab}((\textit{Spaces}/X)_\etale)$, then \begin{enumerate} \item $i_f^{-1}\mathcal{I}$ is injective in $\textit{Ab}(Y_\etale)$, \item $\mathcal{I}|_{X_\etale}$ is injective in $\textit{Ab}(X_\etale)$, \end{enumerate} \item If $\mathcal{I}^\bullet$ is a K-injective complex in $\textit{Ab}((\textit{Spaces}/X)_\etale)$, then \begin{enumerate} \item $i_f^{-1}\mathcal{I}^\bullet$ is a K-injective complex in $\textit{Ab}(Y_\etale)$, \item $\mathcal{I}^\bullet|_{X_\etale}$ is a K-injective complex in $\textit{Ab}(X_\etale)$, \end{enumerate} \end{enumerate} The corresponding statements for modules do not hold. \end{lemma} \begin{proof} Parts (1)(b) and (2)(b) follow formally from the fact that the restriction functor $\pi_{X, *} = i_X^{-1}$ is a right adjoint of the exact functor $\pi_X^{-1}$, see Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives} and Derived Categories, Lemma \ref{derived-lemma-adjoint-preserve-K-injectives}. \medskip\noindent Parts (1)(a) and (2)(a) can be seen in two ways. First proof: We can use that $i_f^{-1}$ is a right adjoint of the exact functor $i_{f, !}$. This functor is constructed in Topologies, Lemma \ref{topologies-lemma-put-in-T-etale} for sheaves of sets and for abelian sheaves in Modules on Sites, Lemma \ref{sites-modules-lemma-g-shriek-adjoint}. It is shown in Modules on Sites, Lemma \ref{sites-modules-lemma-exactness-lower-shriek} that it is exact. Second proof. We can use that $i_f = i_Y \circ f_{big}$ as is shown in Topologies, Lemma \ref{topologies-lemma-morphism-big-small-etale}. Since $f_{big}$ is a localization, we see that pullback by it preserves injectives and K-injectives, see Cohomology on Sites, Lemmas \ref{sites-cohomology-lemma-cohomology-of-open} and \ref{sites-cohomology-lemma-restrict-K-injective-to-open}. Then we apply the already proved parts (1)(b) and (2)(b) to the functor $i_Y^{-1}$ to conclude. \medskip\noindent To see a counter example for the case of modules we refer to \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-compare-injectives}. \end{proof} \noindent Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. The commutative diagram of Topologies on Spaces, Lemma \ref{spaces-topologies-lemma-morphism-big-small-etale} (3) leads to a commutative diagram of ringed sites $$\xymatrix{ (Y_{spaces, \etale}, \mathcal{O}_Y) \ar[d]_{f_{spaces, \etale}} & ((\textit{Spaces}/Y)_\etale, \mathcal{O}) \ar[d]^{f_{big}} \ar[l]^{\pi_Y} \\ (X_{spaces, \etale}, \mathcal{O}_X) & ((\textit{Spaces}/X)_\etale, \mathcal{O}) \ar[l]_{\pi_X} }$$ as one easily sees by writing out the definitions of $f_{small}^\sharp$, $f_{big}^\sharp$, $\pi_X^\sharp$, and $\pi_Y^\sharp$. In particular this means that \begin{equation} \label{equation-compare-big-small} (f_{big, *}\mathcal{F})|_{X_\etale} = f_{small, *}(\mathcal{F}|_{Y_\etale}) \end{equation} for any sheaf $\mathcal{F}$ on $(\textit{Spaces}/Y)_\etale$ and if $\mathcal{F}$ is a sheaf of $\mathcal{O}$-modules, then (\ref{equation-compare-big-small}) is an isomorphism of $\mathcal{O}_X$-modules on $X_\etale$. \begin{lemma} \label{lemma-compare-higher-direct-image} Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. \begin{enumerate} \item For $K$ in $D((\textit{Spaces}/Y)_\etale)$ we have $(Rf_{big, *}K)|_{X_\etale} = Rf_{small, *}(K|_{Y_\etale})$ in $D(X_\etale)$. \item For $K$ in $D((\textit{Spaces}/Y)_\etale, \mathcal{O})$ we have $(Rf_{big, *}K)|_{X_\etale} = Rf_{small, *}(K|_{Y_\etale})$ in $D(\textit{Mod}(X_\etale, \mathcal{O}_X))$. \end{enumerate} More generally, let $g : X' \to X$ be an object of $(\textit{Spaces}/X)_\etale$. Consider the fibre product $$\xymatrix{ Y' \ar[r]_{g'} \ar[d]_{f'} & Y \ar[d]^f \\ X' \ar[r]^g & X }$$ Then \begin{enumerate} \item[(3)] For $K$ in $D((\textit{Spaces}/Y)_\etale)$ we have $i_g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K)$ in $D(X'_\etale)$. \item[(4)] For $K$ in $D((\textit{Spaces}/Y)_\etale, \mathcal{O})$ we have $i_g^*(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^*K)$ in $D(\textit{Mod}(X'_\etale, \mathcal{O}_{X'}))$. \item[(5)] For $K$ in $D((\textit{Spaces}/Y)_\etale)$ we have $g_{big}^{-1}(Rf_{big, *}K) = Rf'_{small, *}((g'_{big})^{-1}K)$ in $D((\textit{Spaces}/X')_\etale)$. \item[(6)] For $K$ in $D((\textit{Spaces}/Y)_\etale, \mathcal{O})$ we have $g_{big}^*(Rf_{big, *}K) = Rf'_{small, *}((g'_{big})^*K)$ in $D(\textit{Mod}(X'_\etale, \mathcal{O}_{X'}))$. \end{enumerate} \end{lemma} \begin{proof} Part (1) follows from Lemma \ref{lemma-compare-injectives} and (\ref{equation-compare-big-small}) on choosing a K-injective complex of abelian sheaves representing $K$. \medskip\noindent Part (3) follows from Lemma \ref{lemma-compare-injectives} and Topologies, Lemma \ref{topologies-lemma-morphism-big-small-cartesian-diagram-etale} on choosing a K-injective complex of abelian sheaves representing $K$. \medskip\noindent Part (5) follows from Cohomology on Sites, Lemmas \ref{sites-cohomology-lemma-cohomology-of-open} and \ref{sites-cohomology-lemma-restrict-K-injective-to-open} and Topologies, Lemma \ref{topologies-lemma-morphism-big-small-cartesian-diagram-etale} on choosing a K-injective complex of abelian sheaves representing $K$. \medskip\noindent Part (6): Observe that $g_{big}$ and $g'_{big}$ are localizations and hence $g_{big}^{-1} = g_{big}^*$ and $(g'_{big})^{-1} = (g'_{big})^*$ are the restriction functors. Hence (6) follows from Cohomology on Sites, Lemmas \ref{sites-cohomology-lemma-cohomology-of-open} and \ref{sites-cohomology-lemma-restrict-K-injective-to-open} and Topologies, Lemma \ref{topologies-lemma-morphism-big-small-cartesian-diagram-etale} on choosing a K-injective complex of modules representing $K$. \medskip\noindent Part (2) can be proved as follows. Above we have seen that $\pi_X \circ f_{big} = f_{small} \circ \pi_Y$ as morphisms of ringed sites. Hence we obtain $R\pi_{X, *} \circ Rf_{big, *} = Rf_{small, *} \circ R\pi_{Y, *}$ by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-derived-pushforward-composition}. Since the restriction functors $\pi_{X, *}$ and $\pi_{Y, *}$ are exact, we conclude. \medskip\noindent Part (4) follows from part (6) and part (2) applied to $f' : Y' \to X'$. \end{proof} \noindent Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{H}$ be an abelian sheaf on $(\textit{Spaces}/X)_\etale$. Recall that $H^n_\etale(U, \mathcal{H})$ denotes the cohomology of $\mathcal{H}$ over an object $U$ of $(\textit{Spaces}/X)_\etale$. \begin{lemma} \label{lemma-compare-cohomology} Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Then \begin{enumerate} \item For $K$ in $D(X_\etale)$ we have $H^n_\etale(X, \pi_X^{-1}K) = H^n(X_\etale, K)$. \item For $K$ in $D(X_\etale, \mathcal{O}_X)$ we have $H^n_\etale(X, L\pi_X^*K) = H^n(X_\etale, K)$. \item For $K$ in $D(X_\etale)$ we have $H^n_\etale(Y, \pi_X^{-1}K) = H^n(Y_\etale, f_{small}^{-1}K)$. \item For $K$ in $D(X_\etale, \mathcal{O}_X)$ we have $H^n_\etale(Y, L\pi_X^*K) = H^n(Y_\etale, Lf_{small}^*K)$. \item For $M$ in $D((\textit{Spaces}/X)_\etale)$ we have $H^n_\etale(Y, M) = H^n(Y_\etale, i_f^{-1}M)$. \item For $M$ in $D((\textit{Spaces}/X)_\etale, \mathcal{O})$ we have $H^n_\etale(Y, M) = H^n(Y_\etale, i_f^*M)$. \end{enumerate} \end{lemma} \begin{proof} To prove (5) represent $M$ by a K-injective complex of abelian sheaves and apply Lemma \ref{lemma-compare-injectives} and work out the definitions. Part (3) follows from this as $i_f^{-1}\pi_X^{-1} = f_{small}^{-1}$. Part (1) is a special case of (3). \medskip\noindent Part (6) follows from the very general Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-pullback-same-cohomology}. Then part (4) follows because $Lf_{small}^* = i_f^* \circ L\pi_X^*$. Part (2) is a special case of (4). \end{proof} \begin{lemma} \label{lemma-cohomological-descent-etale} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $K \in D(X_\etale)$ the map $$K \longrightarrow R\pi_{X, *}\pi_X^{-1}K$$ is an isomorphism where $\pi_X : \Sh((\textit{Spaces}/X)_\etale) \to \Sh(X_\etale)$ is as above. \end{lemma} \begin{proof} This is true because both $\pi_X^{-1}$ and $\pi_{X, *} = i_X^{-1}$ are exact functors and the composition $\pi_{X, *} \circ \pi_X^{-1}$ is the identity functor. \end{proof} \begin{lemma} \label{lemma-compare-higher-direct-image-proper} Let $S$ be a scheme. Let $f : Y \to X$ be a proper morphism of algebraic spaces over $S$. Then we have \begin{enumerate} \item $\pi_X^{-1} \circ f_{small, *} = f_{big, *} \circ \pi_Y^{-1}$ as functors $\Sh(Y_\etale) \to \Sh((\textit{Spaces}/X)_\etale)$, \item $\pi_X^{-1}Rf_{small, *}K = Rf_{big, *}\pi_Y^{-1}K$ for $K$ in $D^+(Y_\etale)$ whose cohomology sheaves are torsion, and \item $\pi_X^{-1}Rf_{small, *}K = Rf_{big, *}\pi_Y^{-1}K$ for all $K$ in $D(Y_\etale)$ if $f$ is finite. \end{enumerate} \end{lemma} \begin{proof} Proof of (1). Let $\mathcal{F}$ be a sheaf on $Y_\etale$. Let $g : X' \to X$ be an object of $(\textit{Spaces}/X)_\etale$. Consider the fibre product $$\xymatrix{ Y' \ar[r]_{f'} \ar[d]_{g'} & X' \ar[d]^g \\ Y \ar[r]^f & X }$$ Then we have $$(f_{big, *}\pi_Y^{-1}\mathcal{F})(X') = (\pi_Y^{-1}\mathcal{F})(Y') = ((g'_{small})^{-1}\mathcal{F})(Y') = (f'_{small, *}(g'_{small})^{-1}\mathcal{F})(X')$$ the second equality by Lemma \ref{lemma-describe-pullback}. On the other hand $$(\pi_X^{-1}f_{small, *}\mathcal{F})(X') = (g_{small}^{-1}f_{small, *}\mathcal{F})(X')$$ again by Lemma \ref{lemma-describe-pullback}. Hence by proper base change for sheaves of sets (Lemma \ref{lemma-proper-base-change-f-star}) we conclude the two sets are canonically isomorphic. The isomorphism is compatible with restriction mappings and defines an isomorphism $\pi_X^{-1}f_{small, *}\mathcal{F} = f_{big, *}\pi_Y^{-1}\mathcal{F}$. Thus an isomorphism of functors $\pi_X^{-1} \circ f_{small, *} = f_{big, *} \circ \pi_Y^{-1}$. \medskip\noindent Proof of (2). There is a canonical base change map $\pi_X^{-1}Rf_{small, *}K \to Rf_{big, *}\pi_Y^{-1}K$ for any $K$ in $D(Y_\etale)$, see Cohomology on Sites, Remark \ref{sites-cohomology-remark-base-change}. To prove it is an isomorphism, it suffices to prove the pull back of the base change map by $i_g : \Sh(X'_\etale) \to \Sh((\Sch/X)_\etale)$ is an isomorphism for any object $g : X' \to X$ of $(\Sch/X)_\etale$. Let $T', g', f'$ be as in the previous paragraph. The pullback of the base change map is \begin{align*} g_{small}^{-1}Rf_{small, *}K & = i_g^{-1}\pi_X^{-1}Rf_{small, *}K \\ & \to i_g^{-1}Rf_{big, *}\pi_Y^{-1}K \\ & = Rf'_{small, *}(i_{g'}^{-1}\pi_Y^{-1}K) \\ & = Rf'_{small, *}((g'_{small})^{-1}K) \end{align*} where we have used $\pi_X \circ i_g = g_{small}$, $\pi_Y \circ i_{g'} = g'_{small}$, and Lemma \ref{lemma-compare-higher-direct-image}. This map is an isomorphism by the proper base change theorem (Lemma \ref{lemma-proper-base-change}) provided $K$ is bounded below and the cohomology sheaves of $K$ are torsion. \medskip\noindent Proof of (3). If $f$ is finite, then the functors $f_{small, *}$ and $f_{big, *}$ are exact. This follows from Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-finite-higher-direct-image-zero} for $f_{small}$. Since any base change $f'$ of $f$ is finite too, we conclude from Lemma \ref{lemma-compare-higher-direct-image} part (3) that $f_{big, *}$ is exact too (as the higher derived functors are zero). Thus this case follows from part (1). \end{proof} \section{Comparing fppf and \'etale topologies} \label{section-fppf-etale} \noindent This section is the analogue of \'Etale Cohomology, Section \ref{etale-cohomology-section-fppf-etale}. \medskip\noindent Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. On the category $\textit{Spaces}/X$ we consider the fppf and \'etale topologies. The identity functor $(\textit{Spaces}/X)_\etale \to (\textit{Spaces}/X)_{fppf}$ is continuous and defines a morphism of sites $$\epsilon_X : (\textit{Spaces}/X)_{fppf} \longrightarrow (\textit{Spaces}/X)_\etale$$ by an application of Sites, Proposition \ref{sites-proposition-get-morphism}. Please note that $\epsilon_{X, *}$ is the identity functor on underlying presheaves and that $\epsilon_X^{-1}$ associates to an \'etale sheaf the fppf sheafification. Consider the morphism of sites $$\pi_X : (\textit{Spaces}/X)_\etale \longrightarrow X_{spaces, \etale}$$ comparing big and small \'etale sites, see Section \ref{section-compare}. The composition determines a morphism of sites $$a_X = \pi_X \circ \epsilon_X : (\textit{Spaces}/X)_{fppf} \longrightarrow X_{spaces, \etale}$$ If $\mathcal{H}$ is an abelian sheaf on $(\textit{Spaces}/X)_{fppf}$, then we will write $H^n_{fppf}(U, \mathcal{H})$ for the cohomology of $\mathcal{H}$ over an object $U$ of $(\textit{Spaces}/X)_{fppf}$. \begin{lemma} \label{lemma-comparison-fppf-etale} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item For $\mathcal{F} \in \Sh(X_\etale)$ we have $\epsilon_{X, *}a_X^{-1}\mathcal{F} = \pi_X^{-1}\mathcal{F}$ and $a_{X, *}a_X^{-1}\mathcal{F} = \mathcal{F}$. \item For $\mathcal{F} \in \textit{Ab}(X_\etale)$ we have $R^i\epsilon_{X, *}(a_X^{-1}\mathcal{F}) = 0$ for $i > 0$. \end{enumerate} \end{lemma} \begin{proof} We have $a_X^{-1}\mathcal{F} = \epsilon_X^{-1} \pi_X^{-1}\mathcal{F}$. By Lemma \ref{lemma-describe-pullback} the \'etale sheaf $\pi_X^{-1}\mathcal{F}$ is a sheaf for the fppf topology and therefore is equal to $a_X^{-1}\mathcal{F}$ (as pulling back by $\epsilon_X$ is given by fppf 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 \ref{lemma-describe-pullback}. \medskip\noindent We will prove part (2) by reducing it to the case of schemes -- see part (1) of \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-V-C-all-n-etale-fppf}. This will clearly work'' as every algebraic space is \'etale locally a scheme. The details are given below but we urge the reader to skip the proof. \medskip\noindent For an abelian sheaf $\mathcal{H}$ on $(\textit{Spaces}/X)_{fppf}$ the higher direct image $R^p\epsilon_{X, *}\mathcal{H}$ is the sheaf associated to the presheaf $U \mapsto H^p_{fppf}(U, \mathcal{H})$ on $(\textit{Spaces}/X)_\etale$. See Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-higher-direct-images}. Since every object of $(\textit{Spaces}/X)_\etale$ has a covering by schemes, it suffices to prove that given $U/X$ a scheme and $\xi \in H^p_{fppf}(U, a_X^{-1}\mathcal{F})$ we can find an \'etale covering $\{U_i \to U\}$ such that $\xi$ restricts to zero on $U_i$. We have \begin{align*} H^p_{fppf}(U, a_X^{-1}\mathcal{F}) & = H^p((\textit{Spaces}/U)_{fppf}, (a_X^{-1}\mathcal{F})|_{\textit{Spaces}/U}) \\ & = H^p((\Sch/U)_{fppf}, (a_X^{-1}\mathcal{F})|_{\Sch/U}) \end{align*} where the second identification is Lemma \ref{lemma-compare-cohomology-other-topologies} and the first is a general fact about restriction (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-of-open}). Looking at the first paragraph and the corresponding result in the case of schemes (\'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-describe-pullback-pi-fppf}) we conclude that the sheaf $(a_X^{-1}\mathcal{F})|_{\Sch/U}$ matches the pullback by the schemes version of $a_U$''. Therefore we can find an \'etale covering $\{U_i \to U\}$ such that our class dies in $H^p((\Sch/U_i)_{fppf}, (a_X^{-1}\mathcal{F})|_{\Sch/U_i})$ for each $i$, see \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-V-C-all-n-etale-fppf} (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. \end{proof} \noindent The hard work done in the case of schemes now tells us that \'etale and fppf cohomology agree for sheaves coming from the small \'etale site. \begin{lemma} \label{lemma-cohomological-descent-etale-fppf} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $K \in D^+(X_\etale)$ 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 $a_X : \Sh((\textit{Spaces}/X)_{fppf}) \to \Sh(X_\etale)$ as above. \end{lemma} \begin{proof} We only prove the second statement; the first is easier and proved in exactly the same manner. There is an immediate reduction to the case where $K$ is given by a single abelian sheaf. Namely, represent $K$ by a bounded below complex $\mathcal{F}^\bullet$. 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 \ref{derived-lemma-leray-acyclicity}) applied to $a_X^{-1}\mathcal{F}^\bullet$ and the functor $a_{X, *}$ we conclude. From now on assume $K = \mathcal{F}$. \medskip\noindent By Lemma \ref{lemma-comparison-fppf-etale} 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 \ref{sites-cohomology-lemma-relative-Leray}). By Lemma \ref{lemma-comparison-fppf-etale} 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 \ref{lemma-cohomological-descent-etale} we have $R^j\pi_{X, *}(\pi_X^{-1}\mathcal{F}) = 0$ for $j > 0$. This concludes the proof. \end{proof} \begin{lemma} \label{lemma-compare-cohomology-etale-fppf} Let $S$ be a scheme and let $X$ be an algebraic space over $S$. With $a_X : \Sh((\textit{Spaces}/X)_{fppf}) \to \Sh(X_\etale)$ as above: \begin{enumerate} \item $H^q(X_\etale, \mathcal{F}) = H^q_{fppf}(X, a_X^{-1}\mathcal{F})$ for an abelian sheaf $\mathcal{F}$ on $X_\etale$, \item $H^q(X_\etale, K) = H^q_{fppf}(X, a_X^{-1}K)$ for $K \in D^+(X_\etale)$. \end{enumerate} Example: if $A$ is an abelian group, then $H^q_\etale(X, \underline{A}) = H^q_{fppf}(X, \underline{A})$. \end{lemma} \begin{proof} This follows from Lemma \ref{lemma-cohomological-descent-etale-fppf} by Cohomology on Sites, Remark \ref{sites-cohomology-remark-before-Leray}. \end{proof} \begin{lemma} \label{lemma-push-pull-fppf-etale} 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 $$\xymatrix{ \Sh((\textit{Spaces}/X)_{fppf}) \ar[rr]_{f_{big, fppf}} \ar[d]_{\epsilon_X} & & \Sh((\textit{Spaces}/Y)_{fppf}) \ar[d]^{\epsilon_Y} \\ \Sh((\textit{Spaces}/X)_\etale) \ar[rr]^{f_{big, \etale}} & & \Sh((\textit{Spaces}/Y)_\etale) }$$ and $$\xymatrix{ \Sh((\textit{Spaces}/X)_{fppf}) \ar[rr]_{f_{big, fppf}} \ar[d]_{a_X} & & \Sh((\textit{Spaces}/Y)_{fppf}) \ar[d]^{a_Y} \\ \Sh(X_\etale) \ar[rr]^{f_{small}} & & \Sh(Y_\etale) }$$ with $a_X = \pi_X \circ \epsilon_X$ and $a_Y = \pi_X \circ \epsilon_X$. \end{lemma} \begin{proof} This follows immediately from working out the definitions of the morphisms involved, see Topologies on Spaces, Section \ref{spaces-topologies-section-fppf} and Section \ref{section-compare}. \end{proof} \begin{lemma} \label{lemma-proper-push-pull-fppf-etale} In Lemma \ref{lemma-push-pull-fppf-etale} if $f$ is proper, then we have \begin{enumerate} \item $a_Y^{-1} \circ f_{small, *} = f_{big, fppf, *} \circ a_X^{-1}$, and \item $a_Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_X^{-1}K)$ for $K$ in $D^+(X_\etale)$ with torsion cohomology sheaves. \end{enumerate} \end{lemma} \begin{proof} Proof of (1). You can prove this by repeating the proof of Lemma \ref{lemma-compare-higher-direct-image-proper} part (1); we will instead deduce the result from this. As $\epsilon_{Y, *}$ is the identity functor on underlying presheaves, it reflects isomorphisms. Lemma \ref{lemma-comparison-fppf-etale} shows that $\epsilon_{Y, *} \circ a_Y^{-1} = \pi_Y^{-1}$ and similarly for $X$. To show that the canonical map $a_Y^{-1}f_{small, *}\mathcal{F} \to f_{big, fppf, *}a_X^{-1}\mathcal{F}$ 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, fppf, *}a_X^{-1}\mathcal{F} \\ & = f_{big, \etale, *} \epsilon_{X, *}a_X^{-1}\mathcal{F} \\ & = f_{big, \etale, *}\pi_X^{-1}\mathcal{F} \end{align*} is an isomorphism. This is part (1) of Lemma \ref{lemma-compare-higher-direct-image-proper}. \medskip\noindent To see (2) we use that \begin{align*} R\epsilon_{Y, *}Rf_{big, fppf, *}a_X^{-1}K & = Rf_{big, \etale, *}R\epsilon_{X, *}a_X^{-1}K \\ & = Rf_{big, \etale, *}\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 \ref{lemma-push-pull-fppf-etale} and Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-derived-pushforward-composition}. Then second equality is Lemma \ref{lemma-cohomological-descent-etale-fppf}. The third is Lemma \ref{lemma-compare-higher-direct-image-proper} part (2). The fourth is Lemma \ref{lemma-cohomological-descent-etale-fppf} again. Thus the base change map $a_Y^{-1}(Rf_{small, *}K) \to Rf_{big, fppf, *}(a_X^{-1}K)$ induces an isomorphism $$R\epsilon_{Y, *}a_Y^{-1}Rf_{small, *}K \to R\epsilon_{Y, *}Rf_{big, fppf, *}a_X^{-1}K$$ The proof is finished by the following remark: a map $\alpha : a_Y^{-1}L \to M$ with $L$ in $D^+(Y_\etale)$ and $M$ in $D^+((\textit{Spaces}/Y)_{fppf})$ such that $R\epsilon_{Y, *}\alpha$ is an isomorphism, 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 \ref{lemma-comparison-fppf-etale} 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. \end{proof} \begin{lemma} \label{lemma-finite-push-pull-fppf-etale} In Lemma \ref{lemma-push-pull-fppf-etale} if $f$ is finite, then $a_Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_X^{-1}K)$ for $K$ in $D^+(X_\etale)$. \end{lemma} \begin{proof} Let $V \to Y$ be a surjective \'etale morphism where $V$ is a scheme. It suffices to prove the base change map is an isomorphism after restricting to $V$. Hence we may assume that $Y$ is a scheme. As the morphism is finite, hence representable, we conclude that we may assume both $X$ and $Y$ are schemes. In this case the result follows from the case of schemes (\'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-V-C-all-n-etale-fppf} part (2)) using the comparison of topoi discussed in Section \ref{section-api} and in particular given in Lemma \ref{lemma-compare-cohomology-other-topologies}. Some details omitted. \end{proof} \begin{lemma} \label{lemma-descent-sheaf-fppf-etale} In Lemma \ref{lemma-push-pull-fppf-etale} assume $f$ is flat, locally of finite presentation, and surjective. Then the functor $$\Sh(Y_\etale) \longrightarrow \left\{ (\mathcal{G}, \mathcal{H}, \alpha) \middle| \begin{matrix} \mathcal{G} \in \Sh(X_\etale),\ \mathcal{H} \in \Sh((\Sch/Y)_{fppf}), \\ \alpha : a_X^{-1}\mathcal{G} \to f_{big, fppf}^{-1}\mathcal{H} \text{ an isomorphism} \end{matrix} \right\}$$ sending $\mathcal{F}$ to $(f_{small}^{-1}\mathcal{F}, a_Y^{-1}\mathcal{F}, can)$ is an equivalence. \end{lemma} \begin{proof} The functor $a_X^{-1}$ is fully faithful (as $a_{X, *}a_X^{-1} = \text{id}$ by Lemma \ref{lemma-comparison-fppf-etale}). Hence the forgetful functor $(\mathcal{G}, \mathcal{H}, \alpha) \mapsto \mathcal{H}$ identifies the category of triples with a full subcategory of $\Sh((\Sch/Y)_{fppf})$. Moreover, the functor $a_Y^{-1}$ is fully faithful, hence the functor in the lemma is fully faithful as well. \medskip\noindent Suppose that we have an \'etale covering $\{Y_i \to Y\}$. Let $f_i : X_i \to Y_i$ be the base change of $f$. Denote $f_{ij} = f_i \times f_j : X_i \times_X X_j \to Y_i \times_Y Y_j$. Claim: if the lemma is true for $f_i$ and $f_{ij}$ for all $i, j$, then the lemma is true for $f$. To see this, note that the given \'etale covering determines an \'etale covering of the final object in each of the four sites $Y_\etale, X_\etale, (\Sch/Y)_{fppf}, (\Sch/X)_{fppf}$. Thus the category of sheaves is equivalent to the category of glueing data for this covering (Sites, Lemma \ref{sites-lemma-mapping-property-glue}) in each of the four cases. A huge commutative diagram of categories then finishes the proof of the claim. We omit the details. The claim shows that we may work \'etale locally on $Y$. In particular, we may assume $Y$ is a scheme. \medskip\noindent Assume $Y$ is a scheme. Choose a scheme $X'$ and a surjective \'etale morphism $s : X' \to X$. Set $f' = f \circ s : X' \to Y$ and observe that $f'$ is surjective, locally of finite presentation, and flat. Claim: if the lemma is true for $f'$, then it is true for $f$. Namely, given a triple $(\mathcal{G}, \mathcal{H}, \alpha)$ for $f$, we can pullback by $s$ to get a triple $(s_{small}^{-1}\mathcal{G}, \mathcal{H}, s_{big, fppf}^{-1}\alpha)$ for $f'$. A solution for this triple gives a sheaf $\mathcal{F}$ on $Y_\etale$ with $a_Y^{-1}\mathcal{F} = \mathcal{H}$. By the first paragraph of the proof this means the triple is in the essential image. This reduces us to the case where both $X$ and $Y$ are schemes. This case follows from \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-descent-sheaf-fppf-etale} via the discussion in Section \ref{section-api} and in particular Lemma \ref{lemma-compare-cohomology-other-topologies}. \end{proof} \section{Comparing fppf and \'etale topologies: modules} \label{section-fppf-etale-modules} \noindent We continue the discussion in Section \ref{section-fppf-etale} but in this section we briefly discuss what happens for sheaves of modules. \medskip\noindent Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The morphisms of sites $\epsilon_X$, $\pi_X$, and their composition $a_X$ introduced in Section \ref{section-fppf-etale} have natural enhancements to morphisms of ringed sites. The first is written as $$\epsilon_X : ((\textit{Spaces}/X)_{fppf}, \mathcal{O}) \longrightarrow ((\textit{Spaces}/X)_\etale, \mathcal{O})$$ Note that we can use the same symbol for the structure sheaf as indeed the sheaves have the same underlying presheaf. The second is $$\pi_X : ((\textit{Spaces}/X)_\etale, \mathcal{O}) \longrightarrow (X_\etale, \mathcal{O}_X)$$ The third is the morphism $$a_X : ((\textit{Spaces}/X)_{fppf}, \mathcal{O}) \longrightarrow (X_\etale, \mathcal{O}_X)$$ Let us review what we already know about quasi-coherent modules on these sites. \begin{lemma} \label{lemma-review-quasi-coherent} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. \begin{enumerate} \item The rule $$\mathcal{F}^a : (\textit{Spaces}/X)_\etale \longrightarrow \textit{Ab},\quad (f : Y \to X) \longmapsto \Gamma(Y, f^*\mathcal{F})$$ satisfies the sheaf condition for fpqc and a fortiori fppf and \'etale coverings, \item $\mathcal{F}^a = \pi_X^*\mathcal{F}$ on $(\textit{Spaces}/X)_\etale$, \item $\mathcal{F}^a = a_X^*\mathcal{F}$ on $(\textit{Spaces}/X)_{fppf}$, \item the rule $\mathcal{F} \mapsto \mathcal{F}^a$ defines an equivalence between quasi-coherent $\mathcal{O}_X$-modules and quasi-coherent modules on $((\textit{Spaces}/X)_\etale, \mathcal{O})$, \item the rule $\mathcal{F} \mapsto \mathcal{F}^a$ defines an equivalence between quasi-coherent $\mathcal{O}_X$-modules and quasi-coherent modules on $((\textit{Spaces}/X)_{fppf}, \mathcal{O})$, \item we have $\epsilon_{X, *}a_X^*\mathcal{F} = \pi_X^*\mathcal{F}$ and $a_{X, *}a_X^*\mathcal{F} = \mathcal{F}$, \item we have $R^i\epsilon_{X, *}(a_X^*\mathcal{F}) = 0$ and $R^ia_{X, *}(a_X^*\mathcal{F}) = 0$ for $i > 0$. \end{enumerate} \end{lemma} \begin{proof} Part (1) is a consequence of fppf descent of quasi-coherent modules. Namely, suppose that $\{f_i : U_i \to U\}$ is an fpqc covering in $(\textit{Spaces}/X)_\etale$. Denote $g : U \to X$ the structure morphism. Suppose that we have a family of sections $s_i \in \Gamma(U_i , f_i^*g^*\mathcal{F})$ such that $s_i|_{U_i \times_U U_j} = s_j|_{U_i \times_U U_j}$. We have to find the correspond section $s \in \Gamma(U, g^*\mathcal{F})$. We can reinterpret the $s_i$ as a family of maps $\varphi_i : f_i^*\mathcal{O}_U = \mathcal{O}_{U_i} \to f_i^*g^*\mathcal{F}$ compatible with the canonical descent data associated to the quasi-coherent sheaves $\mathcal{O}_U$ and $g^*\mathcal{F}$ on $U$. Hence by Descent on Spaces, Proposition \ref{spaces-descent-proposition-fpqc-descent-quasi-coherent} we see that we may (uniquely) descend these to a map $\mathcal{O}_U \to g^*\mathcal{F}$ which gives us our section $s$. \medskip\noindent We will deduce (2) -- (7) from the corresponding statement for schemes. Choose an \'etale covering $\{X_i \to X\}_{i \in I}$ where each $X_i$ is a scheme. Observe that $X_i \times_X X_j$ is a scheme too. This covering induces a covering of the final object in each of the three sites $(\textit{Spaces}/X)_{fppf}$, $(\textit{Spaces}/X)_\etale$, and $X_\etale$. Hence we see that the category of sheaves on these sites are equivalent to descent data for these coverings, see Sites, Lemma \ref{sites-lemma-mapping-property-glue}. Parts (2), (3) are local (because we have the glueing statement). Being quasi-coherent is a local property, hence parts (4), (5) are local. Clearly (6) and (7) are local. It follows that it suffices to prove parts (2) -- (7) of the lemma when $X$ is a scheme. \medskip\noindent Assume $X$ is a scheme. The embeddings $(\Sch/X)_\etale \subset (\textit{Spaces}/X)_\etale$ and $(\Sch/X)_{fppf} \subset (\textit{Spaces}/X)_{fppf}$ determine equivalences of ringed topoi by Lemma \ref{lemma-compare-cohomology-other-topologies}. We conclude that (2) -- (7) follows from the case of schemes. \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-review-quasi-coherent}. To transport the property of being quasi-coherent via this equivalence use that being quasi-coherent is an intrinsic property of modules as explained in Modules on Sites, Section \ref{sites-modules-section-local}. Some minor details omitted. \end{proof} \begin{lemma} \label{lemma-cohomological-descent-etale-fppf-modules} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $\mathcal{F}$ a quasi-coherent $\mathcal{O}_X$-module the maps $$\pi_X^*\mathcal{F} \longrightarrow R\epsilon_{X, *}(a_X^*\mathcal{F}) \quad\text{and}\quad \mathcal{F} \longrightarrow Ra_{X, *}(a_X^*\mathcal{F})$$ are isomorphisms. \end{lemma} \begin{proof} This is an immediate consequence of parts (6) and (7) of Lemma \ref{lemma-review-quasi-coherent}. \end{proof} \begin{lemma} \label{lemma-vanishing-adequate} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3$ be a complex of quasi-coherent $\mathcal{O}_X$-modules. Set $$\mathcal{H}_\etale = \Ker(\pi_X^*\mathcal{F}_2 \to \pi_X^*\mathcal{F}_3)/ \Im(\pi_X^*\mathcal{F}_1 \to \pi_X^*\mathcal{F}_2)$$ on $(\textit{Spaces}/X)_\etale$ and set $$\mathcal{H}_{fppf} = \Ker(a_X^*\mathcal{F}_2 \to a_X^*\mathcal{F}_3)/ \Im(a_X^*\mathcal{F}_1 \to a_X^*\mathcal{F}_2)$$ on $(\textit{Spaces}/X)_{fppf}$. Then $\mathcal{H}_\etale = \epsilon_{X, *}\mathcal{H}_{fppf}$ and $$H^p_\etale(U, \mathcal{H}_\etale) = H^p_{fppf}(U, \mathcal{H}_{fppf}) = 0$$ for $p > 0$ and any affine object $U$ of $(\textit{Spaces}/X)_\etale$. \end{lemma} \noindent More is true, namely the collection of modules on $(\textit{Spaces}/X)_{fppf}$ which fppf locally look like those in the lemma are called adquate modules. They form a weak Serre subcategory of the category of all $\mathcal{O}$-modules and their cohomology is studied in Adequate Modules, Section \ref{adequate-section-adequate}. \begin{proof} For any object $f : U \to X$ of $(\textit{Spaces}/X)_\etale$ consider the restriction $\mathcal{H}_\etale|_{U_\etale}$ of $\mathcal{H}_\etale$ to $U_\etale$ via the functor $i_f^* = i_f^{-1}$ discussed in Section \ref{section-compare}. The sheaf $\mathcal{H}_\etale|_{U_\etale}$ is equal to the homology of complex $f^*\mathcal{F}_\bullet$ in degree $1$. This is true because $i_f \circ \pi_X = f$ as morphisms of ringed sites $U_\etale \to X_\etale$. In particular we see that $\mathcal{H}_\etale|_{U_\etale}$ is a quasi-coherent $\mathcal{O}_U$-module. Next, let $g : V \to U$ be a flat morphism in $(\textit{Spaces}/X)_\etale$. Since $$i_{f \circ g}^* \circ \pi_X^* = (f \circ g)^* = g^* \circ f^*$$ as morphisms of sites $V_\etale \to X_\etale$ and since $g$ is flat hence $g^*$ is exact, we obtain $$\mathcal{H}_\etale|_{V_\etale} = g^*\left(\mathcal{H}_\etale|_{U_\etale}\right)$$ With these preparations we are ready to prove the lemma. \medskip\noindent Let $\mathcal{U} = \{g_i : U_i \to U\}_{i \in I}$ be an fppf covering with $f : U \to X$ as above. The sheaf propery holds for $\mathcal{H}_\etale$ and the covering $\mathcal{U}$ by (1) of Lemma \ref{lemma-review-quasi-coherent} applied to $\mathcal{H}_\etale|_{U_\etale}$ and the above. Therefore we see that $\mathcal{H}_\etale$ is already an fppf sheaf and this means that $\mathcal{H}_{fppf}$ is equal to $\mathcal{H}_\etale$ as a presheaf. In particular $\mathcal{H}_\etale = \epsilon_{X, *}\mathcal{H}_{fppf}$. \medskip\noindent Finally, to prove the vanishing, we use Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cech-vanish-collection}. We let $\mathcal{B}$ be the affine objects of $(\textit{Spaces}/X)_{fppf}$ and we let $\text{Cov}$ be the set of finite fppf coverings $\mathcal{U} = \{U_i \to U\}_{i = 1, \ldots, n}$ with $U$, $U_i$ affine. We have $${\check H}^p(\mathcal{U}, \mathcal{H}_\etale) = {\check H}^p(\mathcal{U}, \left(\mathcal{H}_\etale|_{U_\etale}\right)^a)$$ because the values of $\mathcal{H}_\etale$ on the affine schemes $U_{i_0} \times_U \ldots \times_U U_{i_p}$ flat over $U$ agree with the values of the pullback of the quasi-coherent module $\mathcal{H}_\etale|_{U_\etale}$ by the first paragraph. Hence we obtain vanishing by Descent, Lemma \ref{descent-lemma-standard-covering-Cech-quasi-coherent}. This finishes the proof. \end{proof} \begin{lemma} \label{lemma-cohomological-descent-etale-fppf-modules-unbounded} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $K \in D_\QCoh(\mathcal{O}_X)$ the maps $$L\pi_X^*K \longrightarrow R\epsilon_{X, *}(La_X^*\mathcal{F}) \quad\text{and}\quad K \longrightarrow Ra_{X, *}(La_X^*K)$$ are isomorphisms. Here $a_X : \Sh((\textit{Spaces}/X)_{fppf}) \to \Sh(X_\etale)$ is as above. \end{lemma} \begin{proof} The question is \'etale local on $X$ hence we may assume $X$ is affine. Say $X = \Spec(A)$. Then we have $D_\QCoh(\mathcal{O}_X) = D(A)$ by Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site} and Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-compare-bounded}. Hence we can choose an K-flat complex of $A$-modules $K^\bullet$ whose corresponding complex $\mathcal{K}^\bullet$ of quasi-coherent $\mathcal{O}_X$-modules represents $K$. We claim that $\mathcal{K}^\bullet$ is a K-flat complex of $\mathcal{O}_X$-modules. \medskip\noindent Proof of the claim. By Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-K-flat} we see that $\widetilde{K}^\bullet$ is K-flat on the scheme $(\Spec(A), \mathcal{O}_{\Spec(A)})$. Next, note that $\mathcal{K}^\bullet = \epsilon^*\widetilde{K}^\bullet$ where $\epsilon$ is as in Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site} whence $\mathcal{K}^\bullet$ is K-flat by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-pullback-K-flat-points} and the fact that the \'etale site of a scheme has enough points (\'Etale Cohomology, Remarks \ref{etale-cohomology-remarks-enough-points}). \medskip\noindent By the claim we see that $La_X^*K = a_X^*\mathcal{K}^\bullet$ and $L\pi_X^*K = \pi_X^*\mathcal{K}^\bullet$. Since the first part of the proof shows that the pullback $a_X^*\mathcal{K}^n$ of the quasi-coherent module is acyclic for $\epsilon_{X, *}$, resp.\ $a_{X, *}$, surely the proof is done by Leray's acyclicity lemma? Actually..., no because Leray's acyclicity lemma only applies to bounded below complexes. However, in the next paragraph we will show the result does follow from the bounded below case because our complex is the derived limit of bounded below complexes of quasi-coherent modules. \medskip\noindent The cohomology sheaves of $\pi_X^*\mathcal{K}^\bullet$ and $a_X^*\mathcal{K}^\bullet$ have vanishing higher cohomology groups over affine objects of $(\textit{Spaces}/X)_\etale$ by Lemma \ref{lemma-vanishing-adequate}. Therefore we have $$L\pi_X^*K = R\lim \tau_{\geq -n}(L\pi_X^*K) \quad\text{and}\quad La_X^*K = R\lim \tau_{\geq -n}(La_X^*K)$$ by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-is-limit-dimension}. \medskip\noindent Proof of $L\pi_X^*K = R\epsilon_{X, *}(La_X^*\mathcal{F})$. By the above we have $$R\epsilon_{X, *}La_X^*K = R\lim R\epsilon_{X, *}(\tau_{\geq -n}(La_X^*K))$$ by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-Rf-commutes-with-Rlim}. Note that $\tau_{\geq -n}(La_X^*K)$ is represented by $\tau_{\geq -n}(a_X^*\mathcal{K}^\bullet)$ which may not be the same as $a_X^*(\tau_{\geq -n}\mathcal{K}^\bullet)$. But clearly the systems $$\{\tau_{\geq -n}(a_X^*\mathcal{K}^\bullet)\}_{n \geq 1} \quad\text{and}\quad \{a_X^*(\tau_{\geq -n}\mathcal{K}^\bullet)\}_{n \geq 1}$$ are isomorphic as pro-systems. By Leray's acyclicity lemma (Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity}) and the first part of the lemma we see that $$R\epsilon_{X, *}(a_X^*(\tau_{\geq -n}\mathcal{K}^\bullet)) = \pi_X^*(\tau_{\geq -n}\mathcal{K}^\bullet)$$ Then we can use that the systems $$\{\tau_{\geq -n}(\pi_X^*\mathcal{K}^\bullet)\}_{n \geq 1} \quad\text{and}\quad \{\pi_X^*(\tau_{\geq -n}\mathcal{K}^\bullet)\}_{n \geq 1}$$ are isomorphic as pro-systems. Finally, we put everything together as follows \begin{align*} R\epsilon_{X, *}La_X^*K & = R\epsilon_{X, *} (R\lim \tau_{\geq -n}(La_X^*K)) \\ & = R\lim R\epsilon_{X, *}(\tau_{\geq -n}(La_X^*K)) \\ & = R\lim R\epsilon_{X, *}(\tau_{\geq -n}(a_X^*\mathcal{K}^\bullet)) \\ & = R\lim R\epsilon_{X, *}(a_X^*(\tau_{\geq -n}\mathcal{K}^\bullet)) \\ & = R\lim \pi_X^*(\tau_{\geq -n}\mathcal{K}^\bullet) \\ & = R\lim \tau_{\geq -n}(\pi_X^*\mathcal{K}^\bullet) \\ & = R\lim \tau_{\geq -n}(L\pi_X^*K) \\ & = L\pi_X^*K \end{align*} Here in equalities four and six we have used that isomorphic pro-systems have the same $R\lim$ (small detail omitted). You can avoid this step by using more about cohomology of the terms of the complex $\tau_{\geq -n}a_X^*\mathcal{K}^\bullet$ proved in Lemma \ref{lemma-vanishing-adequate} as this will prove directly that $R\epsilon_{X, *}(\tau_{\geq -n}(a_X^*\mathcal{K}^\bullet)) = \tau_{\geq -n}(\pi_X^*\mathcal{K}^\bullet)$. \medskip\noindent The equality $K = Ra_{X, *}(La_X^*\mathcal{F})$ is proved in exactly the same way using in the final step that $K = R\lim \tau_{\geq -n}K$ by Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-nice-K-injective}. \end{proof} \section{Comparing ph and \'etale topologies} \label{section-ph-etale} \noindent This section is the analogue of \'Etale Cohomology, Section \ref{etale-cohomology-section-ph-etale}. \medskip\noindent Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. On the category $\textit{Spaces}/X$ we consider the ph and \'etale topologies. The identity functor $(\textit{Spaces}/X)_\etale \to (\textit{Spaces}/X)_{ph}$ is continuous as every \'etale covering is a ph covering by Topologies on Spaces, Lemma \ref{spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf-ph}. Hence it defines a morphism of sites $$\epsilon_X : (\textit{Spaces}/X)_{ph} \longrightarrow (\textit{Spaces}/X)_\etale$$ by an application of Sites, Proposition \ref{sites-proposition-get-morphism}. Please note that $\epsilon_{X, *}$ is the identity functor on underlying presheaves and that $\epsilon_X^{-1}$ associates to an \'etale sheaf the ph sheafification. Consider the morphism of sites $$\pi_X : (\textit{Spaces}/X)_\etale \longrightarrow X_{spaces, \etale}$$ comparing big and small \'etale sites, see Section \ref{section-compare}. The composition determines a morphism of sites $$a_X = \pi_X \circ \epsilon_X : (\textit{Spaces}/X)_{ph} \longrightarrow X_{spaces, \etale}$$ 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 $(\textit{Spaces}/X)_{ph}$. \begin{lemma} \label{lemma-comparison-ph-etale} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item For $\mathcal{F} \in \Sh(X_\etale)$ we have $\epsilon_{X, *}a_X^{-1}\mathcal{F} = \pi_X^{-1}\mathcal{F}$ and $a_{X, *}a_X^{-1}\mathcal{F} = \mathcal{F}$. \item For $\mathcal{F} \in \textit{Ab}(X_\etale)$ torsion we have $R^i\epsilon_{X, *}(a_X^{-1}\mathcal{F}) = 0$ for $i > 0$. \end{enumerate} \end{lemma} \begin{proof} We have $a_X^{-1}\mathcal{F} = \epsilon_X^{-1} \pi_X^{-1}\mathcal{F}$. By Lemma \ref{lemma-describe-pullback} the \'etale 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 \ref{lemma-describe-pullback}. \medskip\noindent We will prove part (2) by reducing it to the case of schemes -- see part (1) of \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-V-C-all-n-etale-ph}. This will clearly work'' as every algebraic space is \'etale locally a scheme. The details are given below but we urge the reader to skip the proof. \medskip\noindent 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 $(\textit{Spaces}/X)_\etale$. See Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-higher-direct-images}. Since every object of $(\textit{Spaces}/X)_\etale$ 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 \'etale 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((\Sch/U)_{ph}, (a_X^{-1}\mathcal{F})|_{\Sch/U}) \end{align*} where the second identification is Lemma \ref{lemma-compare-cohomology-other-topologies} and the first is a general fact about restriction (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-of-open}). Looking at the first paragraph and the corresponding result in the case of schemes (\'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-describe-pullback-pi-ph}) we conclude that the sheaf $(a_X^{-1}\mathcal{F})|_{\Sch/U}$ matches the pullback by the schemes version of $a_U$''. Therefore we can find an \'etale covering $\{U_i \to U\}$ such that our class dies in $H^p((\Sch/U_i)_{ph}, (a_X^{-1}\mathcal{F})|_{\Sch/U_i})$ for each $i$, see \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-V-C-all-n-etale-ph} (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. \end{proof} \noindent The hard work done in the case of schemes now tells us that \'etale and ph cohomology agree for torsion abelian sheaves coming from the small \'etale site. \begin{lemma} \label{lemma-cohomological-descent-etale-ph} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $K \in D^+(X_\etale)$ 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 $a_X : \Sh((\textit{Spaces}/X)_{ph}) \to \Sh(X_\etale)$ as above. \end{lemma} \begin{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 \ref{sites-cohomology-lemma-torsion}. 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 \ref{derived-lemma-leray-acyclicity}) 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. \medskip\noindent By Lemma \ref{lemma-comparison-ph-etale} 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 \ref{sites-cohomology-lemma-relative-Leray}). By Lemma \ref{lemma-comparison-ph-etale} 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 \ref{lemma-cohomological-descent-etale} we have $R^j\pi_{X, *}(\pi_X^{-1}\mathcal{F}) = 0$ for $j > 0$. This concludes the proof. \end{proof} \begin{lemma} \label{lemma-compare-cohomology-etale-ph} Let $S$ be a scheme and let $X$ be an algebraic space over $S$. With $a_X : \Sh((\textit{Spaces}/X)_{ph}) \to \Sh(X_\etale)$ as above: \begin{enumerate} \item $H^q(X_\etale, \mathcal{F}) = H^q_{ph}(X, a_X^{-1}\mathcal{F})$ for a torsion abelian sheaf $\mathcal{F}$ on $X_\etale$, \item $H^q(X_\etale, K) = H^q_{ph}(X, a_X^{-1}K)$ for $K \in D^+(X_\etale)$ with torsion cohomology sheaves \end{enumerate} Example: if $A$ is a torsion abelian group, then $H^q_\etale(X, \underline{A}) = H^q_{ph}(X, \underline{A})$. \end{lemma} \begin{proof} This follows from Lemma \ref{lemma-cohomological-descent-etale-ph} by Cohomology on Sites, Remark \ref{sites-cohomology-remark-before-Leray}. \end{proof} \begin{lemma} \label{lemma-push-pull-ph-etale} 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 $$\xymatrix{ \Sh((\textit{Spaces}/X)_{ph}) \ar[rr]_{f_{big, ph}} \ar[d]_{\epsilon_X} & & \Sh((\textit{Spaces}/Y)_{ph}) \ar[d]^{\epsilon_Y} \\ \Sh((\textit{Spaces}/X)_\etale) \ar[rr]^{f_{big, \etale}} & & \Sh((\textit{Spaces}/Y)_\etale) }$$ and $$\xymatrix{ \Sh((\textit{Spaces}/X)_{ph}) \ar[rr]_{f_{big, ph}} \ar[d]_{a_X} & & \Sh((\textit{Spaces}/Y)_{ph}) \ar[d]^{a_Y} \\ \Sh(X_\etale) \ar[rr]^{f_{small}} & & \Sh(Y_\etale) }$$ with $a_X = \pi_X \circ \epsilon_X$ and $a_Y = \pi_X \circ \epsilon_X$. \end{lemma} \begin{proof} This follows immediately from working out the definitions of the morphisms involved, see Topologies on Spaces, Section \ref{spaces-topologies-section-ph} and Section \ref{section-compare}. \end{proof} \begin{lemma} \label{lemma-proper-push-pull-ph-etale} In Lemma \ref{lemma-push-pull-ph-etale} if $f$ is proper, then we have \begin{enumerate} \item $a_Y^{-1} \circ f_{small, *} = f_{big, ph, *} \circ a_X^{-1}$, and \item $a_Y^{-1}(Rf_{small, *}K) = Rf_{big, ph, *}(a_X^{-1}K)$ for $K$ in $D^+(X_\etale)$ with torsion cohomology sheaves. \end{enumerate} \end{lemma} \begin{proof} Proof of (1). You can prove this by repeating the proof of Lemma \ref{lemma-compare-higher-direct-image-proper} part (1); we will instead deduce the result from this. As $\epsilon_{Y, *}$ is the identity functor on underlying presheaves, it reflects isomorphisms. Lemma \ref{lemma-comparison-ph-etale} 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, \etale, *} \epsilon_{X, *}a_X^{-1}\mathcal{F} \\ & = f_{big, \etale, *}\pi_X^{-1}\mathcal{F} \end{align*} is an isomorphism. This is part (1) of Lemma \ref{lemma-compare-higher-direct-image-proper}. \medskip\noindent To see (2) we use that \begin{align*} R\epsilon_{Y, *}Rf_{big, ph, *}a_X^{-1}K & = Rf_{big, \etale, *}R\epsilon_{X, *}a_X^{-1}K \\ & = Rf_{big, \etale, *}\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 \ref{lemma-push-pull-ph-etale} and Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-derived-pushforward-composition}. Then second equality is Lemma \ref{lemma-cohomological-descent-etale-ph}. The third is Lemma \ref{lemma-compare-higher-direct-image-proper} part (2). The fourth is Lemma \ref{lemma-cohomological-descent-etale-ph} 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 $D^+(Y_\etale)$ 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 \ref{lemma-comparison-ph-etale} 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. \end{proof} \input{chapters} \bibliography{my} \bibliographystyle{amsalpha} \end{document}