Introduction
In this chapter continues the discussion started in Cohomology of Spaces, Section 071U. One can also view this chapter as the analogue for algebraic spaces of the chapter on étale cohomology for schemes, see Étale Cohomology, Section 03N2.
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 [Kn].
Conventions
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.
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 \(X \times X\).
Transporting results from schemes
In this section we explain briefly how results for schemes imply results for (representable) algebraic spaces and (representable) morphisms of algebraic spaces. For quasi-coherent modules more is true (because étale cohomology of a quasi-coherent module over a scheme agrees with Zariski cohomology) and this has already been discussed in Cohomology of Spaces, Section 071Y.
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Now suppose that \(X\) is representable by the scheme \(X_0\) (awkward but temporary notation; we usually just say “\(X\) is a scheme”). In this case \(X\) and \(X_0\) have the same small étale sites: \[X_\etale = (X_0)_\etale\] This is pointed out in Properties of Spaces, Section 03EB. Moreover, if \(f : X \to Y\) is a morphism of representable algebraic spaces over \(S\) and if \(f_0 : X_0 \to Y_0\) is a morphism of schemes representing \(f\), then the induced morphisms of small étale topoi agree: \[\xymatrix{ \Sh(X_\etale) \ar[rr]_{f_{small}} \ar@{=}[d] & & \Sh(Y_\etale) \ar@{=}[d] \\ \Sh((X_0)_\etale) \ar[rr]^{(f_0)_{small}} & & \Sh((Y_0)_\etale) }\] See Properties of Spaces, Lemma 03G2 and Topologies, Lemma 021I.
Thus there is absolutely no difference between étale cohomology of a scheme and the étale cohomology of the corresponding algebraic space. Similarly for higher direct images along morphisms of schemes. In fact, if \(f : X \to Y\) is a morphism of algebraic spaces over \(S\) which is representable (by schemes), then the higher direct images \(R^if_*\mathcal{F}\) of a sheaf \(\mathcal{F}\) on \(X_\etale\) can be computed étale locally on \(Y\) (Cohomology on Sites, Lemma 072W) hence this often reduces computations and proofs to the case where \(Y\) and \(X\) are schemes.
We will use the above without further mention in this chapter. For other topologies the same thing is true; we state it explicitly as a lemma for cohomology here.
Lemma
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 03CG) and hence identifies topoi.
Proof
The conditions of Sites, Lemma 03A0 are immediately verified as our functor is fully faithful and as every algebraic space has an étale covering by schemes.
Proper base change
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.
Lemma
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 \(g : Y \times_X Y \to X\).
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 04K2 for the last equality.
The second statement is local on \(X\) in the étale topology, hence we may and do assume \(Y\) is an affine scheme.
Choose a surjective proper morphism \(Z \to Y\) where \(Z\) is a scheme, see Cohomology of Spaces, Lemma 089J. 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 0DES) the result for \(Z \to X\) follows from Étale Cohomology, Lemma 0DDW (in fact it is in some sense equivalent to this lemma).
Lemma
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 \(\Gamma(X, \mathcal{F}) = \Gamma(X_0, i^{-1}\mathcal{F})\).
Proof
Choose a surjective proper morphism \(Y \to X\) where \(Y\) is a scheme, see Cohomology of Spaces, Lemma 089J. Consider the diagram \[\xymatrix{ \Gamma(X_0, \mathcal{F}_0) \ar[r] & \Gamma(Y_0, \mathcal{G}_0) \ar@<1ex>[r] \ar@<-1ex>[r] & \Gamma((Y \times_X Y)_0, \mathcal{H}_0) \\ \Gamma(X, \mathcal{F}) \ar[r] \ar[u] & \Gamma(Y, \mathcal{G}) \ar@<1ex>[r] \ar@<-1ex>[r] \ar[u] & \Gamma(Y \times_X Y, \mathcal{H}) \ar[u] }\] 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 (Étale Cohomology, Lemma 0A0C) we see that the middle and right vertical arrows are bijective. By Lemma 0DFX it follows that the left one is too.
Lemma
Let \(A\) be a henselian local ring. Let \(X\) be an algebraic space over \(A\) such that \(f : X \to \Spec(A)\) is 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 \(\Gamma(X, \mathcal{F}) = \Gamma(X_0, \mathcal{F}|_{X_0})\).
Proof
This is a special case of Lemma 0DFY.
Lemma
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 \(g^{-1}f_*\mathcal{F} = f'_*(g')^{-1}\mathcal{F}\).
Proof
The question is étale local on \(Y'\). Choose a scheme \(V\) and a surjective étale morphism \(V \to Y\). Choose a scheme \(V'\) and a surjective étale 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.
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 089J. 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 (Étale Cohomology, Lemma 0A3U) 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 0DFX 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).
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 \(\Gamma(Y_{\overline{x}}, \mathcal{F}_{\overline{x}})\).
Lemma
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.
Proof
This is a special case of Lemma 0DG0.
Theorem
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.
Proof
This proof repeats a few of the arguments given in the proof of the proper base change theorem for schemes. See Étale Cohomology, Section 095S for more details.
The statement is étale 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).
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 07U6 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 \(n\).
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 0DG0. Applying Leray’s acyclicity lemma (Derived Categories, Lemma 015E) we conclude it suffices to prove \(R^pf'_*(g')^{-1}\mathcal{I}^m = 0\) for \(p > 0\) and \(m \in \mathbf{Z}\).
Choose a surjective proper morphism \(h : Z \to X\) where \(Z\) is a scheme, see Cohomology of Spaces, Lemma 089J. 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 0DFX). 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 \(h_*\mathcal{J}\).
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 0DG0. This finishes the proof.
Lemma
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.
Proof
This is a simple consequence of the proper base change theorem (Theorem 0DG2) 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 015J. Some details omitted.
Lemma
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.
For a torsion abelian sheaf \(\mathcal{F}\) on \(X_\etale\) we have \((R^nf_*\mathcal{F})_{\overline{y}} = H^n_\etale(X_{\overline{y}}, \mathcal{F}_{\overline{y}})\).
For \(E \in D^+(X_\etale)\) with torsion cohomology sheaves we have \((R^nf_*E)_{\overline{y}} = H^n_\etale(X_{\overline{y}}, E_{\overline{y}})\).
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 0DG2. The second one is a special case of Lemma 0DG3.
Lemma
Let \(k'/k\) 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 \(q \geq 0\).
Proof
This is a special case of Theorem 0DG2.
Comparing big and small topoi
Let \(S\) be a scheme and let \(X\) be an algebraic space over \(S\). In Topologies on Spaces, Lemma 0DF3 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 0DF2 and 0DF6. In Topologies on Spaces, Remark 0DF9 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 04BT) transforms \(\mathcal{O}\) into \(\mathcal{O}_X\). Similarly, \(i_f^{-1}\) transforms \(\mathcal{O}\) into \(\mathcal{O}_Y\). See Topologies on Spaces, Remark 0DF9. 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 0DF4).
Lemma
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.
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.
To prove that \(\pi_X^{-1}\mathcal{F}\) is a sheaf for the ph topology it suffices by Topologies on Spaces, Lemma 0DFM 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 0DFX.
The case of smooth, syntomic, fppf coverings follows from the case of ph coverings by Topologies on Spaces, Lemma 0DFH.
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 Étale Cohomology, Lemma 09XN. Case II: \(U\) is a scheme. Here we choose surjective étale 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 étale 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 \(\mathcal{U}\).
Lemma
Let \(S\) be a scheme. Let \(Y \to X\) be a morphism of \((\textit{Spaces}/S)_\etale\).
If \(\mathcal{I}\) is injective in \(\textit{Ab}((\textit{Spaces}/X)_\etale)\), then
\(i_f^{-1}\mathcal{I}\) is injective in \(\textit{Ab}(Y_\etale)\),
\(\mathcal{I}|_{X_\etale}\) is injective in \(\textit{Ab}(X_\etale)\),
If \(\mathcal{I}^\bullet\) is a K-injective complex in \(\textit{Ab}((\textit{Spaces}/X)_\etale)\), then
\(i_f^{-1}\mathcal{I}^\bullet\) is a K-injective complex in \(\textit{Ab}(Y_\etale)\),
\(\mathcal{I}^\bullet|_{X_\etale}\) is a K-injective complex in \(\textit{Ab}(X_\etale)\),
The corresponding statements for modules do not hold.
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 015Z and Derived Categories, Lemma 08BJ.
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 021F for sheaves of sets and for abelian sheaves in Modules on Sites, Lemma 04BG. It is shown in Modules on Sites, Lemma 04BH that it is exact. Second proof. We can use that \(i_f = i_Y \circ f_{big}\) as is shown in Topologies, Lemma 021I. Since \(f_{big}\) is a localization, we see that pullback by it preserves injectives and K-injectives, see Cohomology on Sites, Lemmas 03F3 and 08FI. Then we apply the already proved parts (1)(b) and (2)(b) to the functor \(i_Y^{-1}\) to conclude.
To see a counter example for the case of modules we refer to Étale Cohomology, Lemma 0758.
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 0DF6 (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 [0DG9]\[\begin{equation} (f_{big, *}\mathcal{F})|_{X_\etale} = f_{small, *}(\mathcal{F}|_{Y_\etale}) \end{equation}\] for any sheaf \(\mathcal{F}\) on \((\textit{Spaces}/Y)_\etale\) and if \(\mathcal{F}\) is a sheaf of \(\mathcal{O}\)-modules, then (0DG9) is an isomorphism of \(\mathcal{O}_X\)-modules on \(X_\etale\).
Lemma
Let \(S\) be a scheme. Let \(f : Y \to X\) be a morphism of algebraic spaces over \(S\).
For \(K\) in \(D((\textit{Spaces}/Y)_\etale)\) we have \((Rf_{big, *}K)|_{X_\etale} = Rf_{small, *}(K|_{Y_\etale})\) in \(D(X_\etale)\).
For \(K\) in \(D((\textit{Spaces}/Y)_\etale, \mathcal{O})\) we have \((Rf_{big, *}K)|_{X_\etale} = Rf_{small, *}(K|_{Y_\etale})\) in \(D(\textit{Mod}(X_\etale, \mathcal{O}_X))\).
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
For \(K\) in \(D((\textit{Spaces}/Y)_\etale)\) we have \(i_g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K)\) in \(D(X'_\etale)\).
For \(K\) in \(D((\textit{Spaces}/Y)_\etale, \mathcal{O})\) we have \(i_g^*(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^*K)\) in \(D(\textit{Mod}(X'_\etale, \mathcal{O}_{X'}))\).
For \(K\) in \(D((\textit{Spaces}/Y)_\etale)\) we have \(g_{big}^{-1}(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^{-1}K)\) in \(D((\textit{Spaces}/X')_\etale)\).
For \(K\) in \(D((\textit{Spaces}/Y)_\etale, \mathcal{O})\) we have \(g_{big}^*(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^*K)\) in \(D(\textit{Mod}(X'_\etale, \mathcal{O}_{X'}))\).
Proof
Part (1) follows from Lemma 0DG8 and (0DG9) on choosing a K-injective complex of abelian sheaves representing \(K\).
Part (3) follows from Lemma 0DG8 and Topologies, Lemma 0DDA on choosing a K-injective complex of abelian sheaves representing \(K\).
Part (5) is Cohomology on Sites, Lemma 0EZ0.
Part (6) is Cohomology on Sites, Lemma 0FN5.
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 0D6E. Since the restriction functors \(\pi_{X, *}\) and \(\pi_{Y, *}\) are exact, we conclude.
Part (4) follows from part (6) and part (2) applied to \(f' : Y' \to X'\).
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 \((\textit{Spaces}/X)_\etale\).
Lemma
Let \(S\) be a scheme. Let \(f : Y \to X\) be a morphism of algebraic spaces over \(S\). Then
For \(K\) in \(D(X_\etale)\) we have \(H^n_\etale(X, \pi_X^{-1}K) = H^n(X_\etale, K)\).
For \(K\) in \(D(X_\etale, \mathcal{O}_X)\) we have \(H^n_\etale(X, L\pi_X^*K) = H^n(X_\etale, K)\).
For \(K\) in \(D(X_\etale)\) we have \(H^n_\etale(Y, \pi_X^{-1}K) = H^n(Y_\etale, f_{small}^{-1}K)\).
For \(K\) in \(D(X_\etale, \mathcal{O}_X)\) we have \(H^n_\etale(Y, L\pi_X^*K) = H^n(Y_\etale, Lf_{small}^*K)\).
For \(M\) in \(D((\textit{Spaces}/X)_\etale)\) we have \(H^n_\etale(Y, M) = H^n(Y_\etale, i_f^{-1}M)\).
For \(M\) in \(D((\textit{Spaces}/X)_\etale, \mathcal{O})\) we have \(H^n_\etale(Y, M) = H^n(Y_\etale, i_f^*M)\).
Proof
To prove (5) represent \(M\) by a K-injective complex of abelian sheaves and apply Lemma 0DG8 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).
Part (6) follows from the very general Cohomology on Sites, Lemma 0DD8. Then part (4) follows because \(Lf_{small}^* = i_f^* \circ L\pi_X^*\). Part (2) is a special case of (4).
Lemma
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.
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.
Lemma
Let \(S\) be a scheme. Let \(f : Y \to X\) be a proper morphism of algebraic spaces over \(S\). Then we have
\(\pi_X^{-1} \circ f_{small, *} = f_{big, *} \circ \pi_Y^{-1}\) as functors \(\Sh(Y_\etale) \to \Sh((\textit{Spaces}/X)_\etale)\),
\(\pi_X^{-1}Rf_{small, *}K = Rf_{big, *}\pi_Y^{-1}K\) for \(K\) in \(D^+(Y_\etale)\) whose cohomology sheaves are torsion, and
\(\pi_X^{-1}Rf_{small, *}K = Rf_{big, *}\pi_Y^{-1}K\) for all \(K\) in \(D(Y_\etale)\) if \(f\) is finite.
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 0DG7. On the other hand \[(\pi_X^{-1}f_{small, *}\mathcal{F})(X') = (g_{small}^{-1}f_{small, *}\mathcal{F})(X')\] again by Lemma 0DG7. Hence by proper base change for sheaves of sets (Lemma 0DG0) 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 \(\pi_X^{-1} \circ f_{small, *} = f_{big, *} \circ \pi_Y^{-1}\).
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 07A7. 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 0DGA. This map is an isomorphism by the proper base change theorem (Lemma 0DG3) provided \(K\) is bounded below and the cohomology sheaves of \(K\) are torsion.
Proof of (3). If \(f\) is finite, then the functors \(f_{small, *}\) and \(f_{big, *}\) are exact. This follows from Cohomology of Spaces, Lemma 0A4K for \(f_{small}\). Since any base change \(f'\) of \(f\) is finite too, we conclude from Lemma 0DGA part (3) that \(f_{big, *}\) is exact too (as the higher derived functors are zero). Thus this case follows from part (1).
Comparing fppf and étale topologies
This section is the analogue of Étale Cohomology, Section 0DDK.
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). On the category \(\textit{Spaces}/X\) we consider the fppf and étale 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 00X6. Please note that \(\epsilon_{X, *}\) is the identity functor on underlying presheaves and that \(\epsilon_X^{-1}\) associates to an étale sheaf the fppf sheafification. Consider the morphism of sites \[\pi_X : (\textit{Spaces}/X)_\etale \longrightarrow X_{spaces, \etale}\] comparing big and small étale sites, see Section 0DG6. 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 \((\textit{Spaces}/X)_{fppf}\).
Lemma
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\).
For \(\mathcal{F} \in \Sh(X_\etale)\) we have \(\epsilon_{X, *}a_X^{-1}\mathcal{F} = \pi_X^{-1}\mathcal{F}\) and \(a_{X, *}a_X^{-1}\mathcal{F} = \mathcal{F}\).
For \(\mathcal{F} \in \textit{Ab}(X_\etale)\) we have \(R^i\epsilon_{X, *}(a_X^{-1}\mathcal{F}) = 0\) for \(i > 0\).
Proof
We have \(a_X^{-1}\mathcal{F} = \epsilon_X^{-1} \pi_X^{-1}\mathcal{F}\). By Lemma 0DG7 the étale 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 0DG7.
We will prove part (2) by reducing it to the case of schemes – see part (1) of Étale Cohomology, Lemma 0DDS. This will “clearly work” as every algebraic space is étale locally a scheme. The details are given below but we urge the reader to skip the proof.
For an abelian sheaf \(\mathcal{H}\) on \((\textit{Spaces}/X)_{fppf}\) 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 072W. 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 étale 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 0DFV and the first is a general fact about restriction (Cohomology on Sites, Lemma 03F3). Looking at the first paragraph and the corresponding result in the case of schemes (Étale Cohomology, Lemma 0DDL) 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 étale 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 Étale Cohomology, Lemma 0DDS (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.
The hard work done in the case of schemes now tells us that étale and fppf cohomology agree for sheaves coming from the small étale site.
Lemma
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.
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 015E) applied to \(a_X^{-1}\mathcal{F}^\bullet\) and the functor \(a_{X, *}\) we conclude. From now on assume \(K = \mathcal{F}\).
By Lemma 0DGF 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 0734). By Lemma 0DGF 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 0DGC we have \(R^j\pi_{X, *}(\pi_X^{-1}\mathcal{F}) = 0\) for \(j > 0\). This concludes the proof.
Lemma
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:
\(H^q(X_\etale, \mathcal{F}) = H^q_{fppf}(X, a_X^{-1}\mathcal{F})\) for an abelian sheaf \(\mathcal{F}\) on \(X_\etale\),
\(H^q(X_\etale, K) = H^q_{fppf}(X, a_X^{-1}K)\) for \(K \in D^+(X_\etale)\).
Example: if \(A\) is an abelian group, then \(H^q_\etale(X, \underline{A}) = H^q_{fppf}(X, \underline{A})\).
Proof
This follows from Lemma 0DGG by Cohomology on Sites, Remark 08J6.
Lemma
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 \(a_Y = \pi_X \circ \epsilon_X\).
Proof
This follows immediately from working out the definitions of the morphisms involved, see Topologies on Spaces, Section 03Y7 and Section 0DG6.
Lemma
In Lemma 0DGI if \(f\) is proper, then we have
\(a_Y^{-1} \circ f_{small, *} = f_{big, fppf, *} \circ a_X^{-1}\), and
\(a_Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_X^{-1}K)\) for \(K\) in \(D^+(X_\etale)\) with torsion cohomology sheaves.
Proof
Proof of (1). You can prove this by repeating the proof of Lemma 0DGD part (1); we will instead deduce the result from this. As \(\epsilon_{Y, *}\) is the identity functor on underlying presheaves, it reflects isomorphisms. Lemma 0DGF 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 0DGD.
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 0DGI and Cohomology on Sites, Lemma 0D6E. Then second equality is Lemma 0DGG. The third is Lemma 0DGD part (2). The fourth is Lemma 0DGG 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 0DGF 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.
Lemma
In Lemma 0DGI if \(f\) is finite, then \(a_Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_X^{-1}K)\) for \(K\) in \(D^+(X_\etale)\).
Proof
Let \(V \to Y\) be a surjective étale 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 (Étale Cohomology, Lemma 0DDS part (2)) using the comparison of topoi discussed in Section 0DFU and in particular given in Lemma 0DFV. Some details omitted.
Lemma
In Lemma 0DGI 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.
Proof
The functor \(a_X^{-1}\) is fully faithful (as \(a_{X, *}a_X^{-1} = \text{id}\) by Lemma 0DGF). 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.
Suppose that we have an étale 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 étale covering determines an étale 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 04TS) 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 étale locally on \(Y\). In particular, we may assume \(Y\) is a scheme.
Assume \(Y\) is a scheme. Choose a scheme \(X'\) and a surjective étale 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 Étale Cohomology, Lemma 0DEU via the discussion in Section 0DFU and in particular Lemma 0DFV.
Comparing fppf and étale topologies: modules
We continue the discussion in Section 0DGE but in this section we briefly discuss what happens for sheaves of modules.
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 0DGE 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.
Lemma
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module.
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 étale coverings,
\(\mathcal{F}^a = \pi_X^*\mathcal{F}\) on \((\textit{Spaces}/X)_\etale\),
\(\mathcal{F}^a = a_X^*\mathcal{F}\) on \((\textit{Spaces}/X)_{fppf}\),
the rule \(\mathcal{F} \mapsto \mathcal{F}^a\) defines an equivalence between quasi-coherent \(\mathcal{O}_X\)-modules and quasi-coherent modules on \(((\textit{Spaces}/X)_\etale, \mathcal{O})\),
the rule \(\mathcal{F} \mapsto \mathcal{F}^a\) defines an equivalence between quasi-coherent \(\mathcal{O}_X\)-modules and quasi-coherent modules on \(((\textit{Spaces}/X)_{fppf}, \mathcal{O})\),
we have \(\epsilon_{X, *}a_X^*\mathcal{F} = \pi_X^*\mathcal{F}\) and \(a_{X, *}a_X^*\mathcal{F} = \mathcal{F}\),
we have \(R^i\epsilon_{X, *}(a_X^*\mathcal{F}) = 0\) and \(R^ia_{X, *}(a_X^*\mathcal{F}) = 0\) for \(i > 0\).
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 04W8 we see that we may (uniquely) descend these to a map \(\mathcal{O}_U \to g^*\mathcal{F}\) which gives us our section \(s\).
We will deduce (2) – (7) from the corresponding statement for schemes. Choose an étale 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 04TS. 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.
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 0DFV. We conclude that (2) – (7) follows from the case of schemes. Étale Cohomology, Lemma 0DEW. 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 03DK. Some minor details omitted.
Lemma
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.
Proof
This is an immediate consequence of parts (6) and (7) of Lemma 0DGN.
Lemma
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 \((\textit{Spaces}/X)_\etale\).
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 06VF.
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 0DG6. 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.
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 property holds for \(\mathcal{H}_\etale\) and the covering \(\mathcal{U}\) by (1) of Lemma 0DGN 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 \(\mathcal{H}_\etale = \epsilon_{X, *}\mathcal{H}_{fppf}\).
Finally, to prove the vanishing, we use Cohomology on Sites, Lemma 03F9. 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 03FJ. This finishes the proof.
Lemma
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). For \(K \in D_\QCoh(\mathcal{O}_X)\) the maps \[L\pi_X^*K \longrightarrow R\epsilon_{X, *}(La_X^*K) \quad\text{and}\quad K \longrightarrow Ra_{X, *}(La_X^*K)\] are isomorphisms. Here \(a_X : \Sh((\textit{Spaces}/X)_{fppf}) \to \Sh(X_\etale)\) is as above.
Proof
The question is étale 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 071Q and Derived Categories of Schemes, Lemma 06Z0. 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.
Proof of the claim. By Derived Categories of Schemes, Lemma 08DV 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 071Q whence \(\mathcal{K}^\bullet\) is K-flat by Cohomology on Sites, Lemma 0DEP and the fact that the étale site of a scheme has enough points (Étale Cohomology, Remarks 040S).
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.
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 0DGQ. 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 0D6P.
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 0A07. 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 015E) 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 0DGQ as this will prove directly that \(R\epsilon_{X, *}(\tau_{\geq -n}(a_X^*\mathcal{K}^\bullet)) = \tau_{\geq -n}(\pi_X^*\mathcal{K}^\bullet)\).
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 08F6.
Comparing ph and étale topologies
This section is the analogue of Étale Cohomology, Section 0DDV.
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). On the category \(\textit{Spaces}/X\) we consider the ph and étale topologies. The identity functor \((\textit{Spaces}/X)_\etale \to (\textit{Spaces}/X)_{ph}\) is continuous as every étale covering is a ph covering by Topologies on Spaces, Lemma 0DFH. Hence it defines a morphism of sites \[\epsilon_X : (\textit{Spaces}/X)_{ph} \longrightarrow (\textit{Spaces}/X)_\etale\] by an application of Sites, Proposition 00X6. Please note that \(\epsilon_{X, *}\) is the identity functor on underlying presheaves and that \(\epsilon_X^{-1}\) associates to an étale sheaf the ph sheafification. Consider the morphism of sites \[\pi_X : (\textit{Spaces}/X)_\etale \longrightarrow X_{spaces, \etale}\] comparing big and small étale sites, see Section 0DG6. 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 \((\textit{Spaces}/X)_{ph}\).
Lemma
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\).
For \(\mathcal{F} \in \Sh(X_\etale)\) we have \(\epsilon_{X, *}a_X^{-1}\mathcal{F} = \pi_X^{-1}\mathcal{F}\) and \(a_{X, *}a_X^{-1}\mathcal{F} = \mathcal{F}\).
For \(\mathcal{F} \in \textit{Ab}(X_\etale)\) torsion we have \(R^i\epsilon_{X, *}(a_X^{-1}\mathcal{F}) = 0\) for \(i > 0\).
Proof
We have \(a_X^{-1}\mathcal{F} = \epsilon_X^{-1} \pi_X^{-1}\mathcal{F}\). By Lemma 0DG7 the étale sheaf \(\pi_X^{-1}\mathcal{F}\) is a sheaf for the ph topology and therefore is equal to \(a_X^{-1}\mathcal{F}\) (as pulling back by \(\epsilon_X\) is given by ph sheafification). Recall moreover that \(\epsilon_{X, *}\) is the identity on underlying presheaves. Now part (1) is immediate from the explicit description of \(\pi_X^{-1}\) in Lemma 0DG7.
We will prove part (2) by reducing it to the case of schemes – see part (1) of Étale Cohomology, Lemma 0DE4. This will “clearly work” as every algebraic space is étale locally a scheme. The details are given below but we urge the reader to skip the proof.
For an abelian sheaf \(\mathcal{H}\) on \((\textit{Spaces}/X)_{ph}\) the higher direct image \(R^p\epsilon_{X, *}\mathcal{H}\) is the sheaf associated to the presheaf \(U \mapsto H^p_{ph}(U, \mathcal{H})\) on \((\textit{Spaces}/X)_\etale\). See Cohomology on Sites, Lemma 072W. 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 étale covering \(\{U_i \to U\}\) such that \(\xi\) restricts to zero on \(U_i\). We have \[\begin{align*} H^p_{ph}(U, a_X^{-1}\mathcal{F}) & = H^p((\textit{Spaces}/U)_{ph}, (a_X^{-1}\mathcal{F})|_{\textit{Spaces}/U}) \\ & = H^p((\Sch/U)_{ph}, (a_X^{-1}\mathcal{F})|_{\Sch/U}) \end{align*}\] where the second identification is Lemma 0DFV and the first is a general fact about restriction (Cohomology on Sites, Lemma 03F3). Looking at the first paragraph and the corresponding result in the case of schemes (Étale Cohomology, Lemma 0DDW) 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 étale 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 Étale Cohomology, Lemma 0DE4 (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.
The hard work done in the case of schemes now tells us that étale and ph cohomology agree for torsion abelian sheaves coming from the small étale site.
Lemma
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.
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 0DD7. 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 015E) 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 0DGT 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 0734). By Lemma 0DGT 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 0DGC we have \(R^j\pi_{X, *}(\pi_X^{-1}\mathcal{F}) = 0\) for \(j > 0\). This concludes the proof.
Lemma
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:
\(H^q(X_\etale, \mathcal{F}) = H^q_{ph}(X, a_X^{-1}\mathcal{F})\) for a torsion abelian sheaf \(\mathcal{F}\) on \(X_\etale\),
\(H^q(X_\etale, K) = H^q_{ph}(X, a_X^{-1}K)\) for \(K \in D^+(X_\etale)\) with torsion cohomology sheaves
Example: if \(A\) is a torsion abelian group, then \(H^q_\etale(X, \underline{A}) = H^q_{ph}(X, \underline{A})\).
Proof
This follows from Lemma 0DGU by Cohomology on Sites, Remark 08J6.
Lemma
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 \(a_Y = \pi_X \circ \epsilon_X\).
Proof
This follows immediately from working out the definitions of the morphisms involved, see Topologies on Spaces, Section 0DFF and Section 0DG6.
Lemma
In Lemma 0DGW if \(f\) is proper, then we have
\(a_Y^{-1} \circ f_{small, *} = f_{big, ph, *} \circ a_X^{-1}\), and
\(a_Y^{-1}(Rf_{small, *}K) = Rf_{big, ph, *}(a_X^{-1}K)\) for \(K\) in \(D^+(X_\etale)\) with torsion cohomology sheaves.
Proof
Proof of (1). You can prove this by repeating the proof of Lemma 0DGD part (1); we will instead deduce the result from this. As \(\epsilon_{Y, *}\) is the identity functor on underlying presheaves, it reflects isomorphisms. Lemma 0DGT 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 0DGD.
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 0DGW and Cohomology on Sites, Lemma 0D6E. Then second equality is Lemma 0DGU. The third is Lemma 0DGD part (2). The fourth is Lemma 0DGU 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 0DGT 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.