96.19 The relative Čech complex
Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ as in (96.18.0.1). Consider the associated simplicial object $\mathcal{U}_\bullet $ and the maps $f_ n : \mathcal{U}_ n \to \mathcal{X}$. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} $. Finally, suppose that $\mathcal{F}$ is a sheaf (of sets) on $\mathcal{X}_\tau $. Then
\[ \xymatrix{ f_{0, *}f_0^{-1}\mathcal{F} \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & f_{1, *}f_1^{-1}\mathcal{F} \ar@<1ex>[r] \ar@<0ex>[r] \ar@<-1ex>[r] & f_{2, *}f_2^{-1}\mathcal{F} } \]
is a cosimplicial sheaf on $\mathcal{X}_\tau $ where we use the pullback maps introduced in Sites, Section 7.45. If $\mathcal{F}$ is an abelian sheaf, then $f_{n, *}f_ n^{-1}\mathcal{F}$ form a cosimplicial abelian sheaf on $\mathcal{X}_\tau $. The associated complex (see Simplicial, Section 14.25)
\[ \ldots \to 0 \to f_{0, *}f_0^{-1}\mathcal{F} \to f_{1, *}f_1^{-1}\mathcal{F} \to f_{2, *}f_2^{-1}\mathcal{F} \to \ldots \]
is called the relative Čech complex associated to the situation. We will denote this complex $\mathcal{K}^\bullet (f, \mathcal{F})$. The extended relative Čech complex is the complex
\[ \ldots \to 0 \to \mathcal{F} \to f_{0, *}f_0^{-1}\mathcal{F} \to f_{1, *}f_1^{-1}\mathcal{F} \to f_{2, *}f_2^{-1}\mathcal{F} \to \ldots \]
with $\mathcal{F}$ in degree $-1$. The extended relative Čech complex is acyclic if and only if the map $\mathcal{F}[0] \to \mathcal{K}^\bullet (f, \mathcal{F})$ is a quasi-isomorphism of complexes of sheaves.
Lemma 96.19.2. Generalities on relative Čech complexes.
If
\[ \xymatrix{ \mathcal{V} \ar[d]_ g \ar[r]_ h & \mathcal{U} \ar[d]^ f \\ \mathcal{Y} \ar[r]^ e & \mathcal{X} } \]
is $2$-commutative diagram of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$, then there is a morphism $e^{-1}\mathcal{K}^\bullet (f, \mathcal{F}) \to \mathcal{K}^\bullet (g, e^{-1}\mathcal{F})$.
if $h$ and $e$ are equivalences, then the map of (1) is an isomorphism,
if $f, f' : \mathcal{U} \to \mathcal{X}$ are $2$-isomorphic, then the associated relative Čech complexes are isomorphic,
Proof.
Literally the same as the proof of Lemma 96.18.1 using the pullback maps of Remark 96.19.1.
$\square$
Lemma 96.19.3. If there exists a $1$-morphism $s : \mathcal{X} \to \mathcal{U}$ such that $f \circ s$ is $2$-isomorphic to $\text{id}_\mathcal {X}$ then the extended relative Čech complex is homotopic to zero.
Proof.
Literally the same as the proof of Lemma 96.18.2.
$\square$
Lemma 96.19.5. Let
\[ \xymatrix{ \mathcal{V} \ar[d]_ g \ar[r]_ h & \mathcal{U} \ar[d]^ f \\ \mathcal{Y} \ar[r]^ e & \mathcal{X} } \]
be a $2$-fibre product of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ and let $\mathcal{F}$ be an abelian presheaf on $\mathcal{X}$. Then the map $e^{-1}\mathcal{K}^\bullet (f, \mathcal{F}) \to \mathcal{K}^\bullet (g, e^{-1}\mathcal{F})$ of Lemma 96.19.2 is an isomorphism of complexes of abelian presheaves.
Proof.
Let $y$ be an object of $\mathcal{Y}$ lying over the scheme $T$. Set $x = e(y)$. We are going to show that the map induces an isomorphism on sections over $y$. Note that
\[ \Gamma (y, e^{-1}\mathcal{K}^\bullet (f, \mathcal{F})) = \Gamma (x, \mathcal{K}^\bullet (f, \mathcal{F})) = \check{\mathcal{C}}^\bullet ( (\mathit{Sch}/T)_{fppf} \times _{x, \mathcal{X}} \mathcal{U} \to (\mathit{Sch}/T)_{fppf}, x^{-1}\mathcal{F}) \]
by Remark 96.19.4. On the other hand,
\[ \Gamma (y, \mathcal{K}^\bullet (g, e^{-1}\mathcal{F})) = \check{\mathcal{C}}^\bullet ( (\mathit{Sch}/T)_{fppf} \times _{y, \mathcal{Y}} \mathcal{V} \to (\mathit{Sch}/T)_{fppf}, y^{-1}e^{-1}\mathcal{F}) \]
also by Remark 96.19.4. Note that $y^{-1}e^{-1}\mathcal{F} = x^{-1}\mathcal{F}$ and since the diagram is $2$-cartesian the $1$-morphism
\[ (\mathit{Sch}/T)_{fppf} \times _{y, \mathcal{Y}} \mathcal{V} \to (\mathit{Sch}/T)_{fppf} \times _{x, \mathcal{X}} \mathcal{U} \]
is an equivalence. Hence the map on sections over $y$ is an isomorphism by Lemma 96.18.1.
$\square$
Exactness can be checked on a “covering”.
Lemma 96.19.6. Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} $. Let
\[ \mathcal{F} \to \mathcal{G} \to \mathcal{H} \]
be a complex in $\textit{Ab}(\mathcal{X}_\tau )$. Assume that
for every object $x$ of $\mathcal{X}$ there exists a covering $\{ x_ i \to x\} $ in $\mathcal{X}_\tau $ such that each $x_ i$ is isomorphic to $f(u_ i)$ for some object $u_ i$ of $\mathcal{U}$, and
$f^{-1}\mathcal{F} \to f^{-1}\mathcal{G} \to f^{-1}\mathcal{H}$ is exact.
Then the sequence $\mathcal{F} \to \mathcal{G} \to \mathcal{H}$ is exact.
Proof.
Let $x$ be an object of $\mathcal{X}$ lying over the scheme $T$. Consider the sequence $x^{-1}\mathcal{F} \to x^{-1}\mathcal{G} \to x^{-1}\mathcal{H}$ of abelian sheaves on $(\mathit{Sch}/T)_\tau $. It suffices to show this sequence is exact. By assumption there exists a $\tau $-covering $\{ T_ i \to T\} $ such that $x|_{T_ i}$ is isomorphic to $f(u_ i)$ for some object $u_ i$ of $\mathcal{U}$ over $T_ i$ and moreover the sequence $u_ i^{-1}f^{-1}\mathcal{F} \to u_ i^{-1}f^{-1}\mathcal{G} \to u_ i^{-1}f^{-1}\mathcal{H}$ of abelian sheaves on $(\mathit{Sch}/T_ i)_\tau $ is exact. Since $u_ i^{-1}f^{-1}\mathcal{F} = x^{-1}\mathcal{F}|_{(\mathit{Sch}/T_ i)_\tau }$ we conclude that the sequence $x^{-1}\mathcal{F} \to x^{-1}\mathcal{G} \to x^{-1}\mathcal{H}$ become exact after localizing at each of the members of a covering, hence the sequence is exact.
$\square$
Proposition 96.19.7. Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} $. If
$\mathcal{F}$ is an abelian sheaf on $\mathcal{X}_\tau $, and
for every object $x$ of $\mathcal{X}$ there exists a covering $\{ x_ i \to x\} $ in $\mathcal{X}_\tau $ such that each $x_ i$ is isomorphic to $f(u_ i)$ for some object $u_ i$ of $\mathcal{U}$,
then the extended relative Čech complex
\[ \ldots \to 0 \to \mathcal{F} \to f_{0, *}f_0^{-1}\mathcal{F} \to f_{1, *}f_1^{-1}\mathcal{F} \to f_{2, *}f_2^{-1}\mathcal{F} \to \ldots \]
is exact in $\textit{Ab}(\mathcal{X}_\tau )$.
Proof.
By Lemma 96.19.6 it suffices to check exactness after pulling back to $\mathcal{U}$. By Lemma 96.19.5 the pullback of the extended relative Čech complex is isomorphic to the extend relative Čech complex for the morphism $\mathcal{U} \times _\mathcal {X} \mathcal{U} \to \mathcal{U}$ and an abelian sheaf on $\mathcal{U}_\tau $. Since there is a section $\Delta _{\mathcal{U}/\mathcal{X}} : \mathcal{U} \to \mathcal{U} \times _\mathcal {X} \mathcal{U}$ exactness follows from Lemma 96.19.3.
$\square$
Using this we can construct the Čech-to-cohomology spectral sequence as follows. We first give a technical, precise version. In the next section we give a version that applies only to algebraic stacks.
Lemma 96.19.8. Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} $. Assume
$\mathcal{F}$ is an abelian sheaf on $\mathcal{X}_\tau $,
for every object $x$ of $\mathcal{X}$ there exists a covering $\{ x_ i \to x\} $ in $\mathcal{X}_\tau $ such that each $x_ i$ is isomorphic to $f(u_ i)$ for some object $u_ i$ of $\mathcal{U}$,
the category $\mathcal{U}$ has equalizers, and
the functor $f$ is faithful.
Then there is a first quadrant spectral sequence of abelian groups
\[ E_1^{p, q} = H^ q((\mathcal{U}_ p)_\tau , f_ p^{-1}\mathcal{F}) \Rightarrow H^{p + q}(\mathcal{X}_\tau , \mathcal{F}) \]
converging to the cohomology of $\mathcal{F}$ in the $\tau $-topology.
Proof.
Before we start the proof we make some remarks. By Lemma 96.17.4 (and induction) all of the categories fibred in groupoids $\mathcal{U}_ p$ have equalizers and all of the morphisms $f_ p : \mathcal{U}_ p \to \mathcal{X}$ are faithful. Let $\mathcal{I}$ be an injective object of $\textit{Ab}(\mathcal{X}_\tau )$. By Lemma 96.17.5 we see $f_ p^{-1}\mathcal{I}$ is an injective object of $\textit{Ab}((\mathcal{U}_ p)_\tau )$. Hence $f_{p, *}f_ p^{-1}\mathcal{I}$ is an injective object of $\textit{Ab}(\mathcal{X}_\tau )$ by Lemma 96.17.1. Hence Proposition 96.19.7 shows that the extended relative Čech complex
\[ \ldots \to 0 \to \mathcal{I} \to f_{0, *}f_0^{-1}\mathcal{I} \to f_{1, *}f_1^{-1}\mathcal{I} \to f_{2, *}f_2^{-1}\mathcal{I} \to \ldots \]
is an exact complex in $\textit{Ab}(\mathcal{X}_\tau )$ all of whose terms are injective. Taking global sections of this complex is exact and we see that the Čech complex $\check{\mathcal{C}}^\bullet (\mathcal{U} \to \mathcal{X}, \mathcal{I})$ is quasi-isomorphic to $\Gamma (\mathcal{X}_\tau , \mathcal{I})[0]$.
With these preliminaries out of the way consider the two spectral sequences associated to the double complex (see Homology, Section 12.25)
\[ \check{\mathcal{C}}^\bullet (\mathcal{U} \to \mathcal{X}, \mathcal{I}^\bullet ) \]
where $\mathcal{F} \to \mathcal{I}^\bullet $ is an injective resolution in $\textit{Ab}(\mathcal{X}_\tau )$. The discussion above shows that Homology, Lemma 12.25.4 applies which shows that $\Gamma (\mathcal{X}_\tau , \mathcal{I}^\bullet )$ is quasi-isomorphic to the total complex associated to the double complex. By our remarks above the complex $f_ p^{-1}\mathcal{I}^\bullet $ is an injective resolution of $f_ p^{-1}\mathcal{F}$. Hence the other spectral sequence is as indicated in the lemma.
$\square$
To be sure there is a version for modules as well.
Lemma 96.19.9. Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} $. Assume
$\mathcal{F}$ is an object of $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$,
for every object $x$ of $\mathcal{X}$ there exists a covering $\{ x_ i \to x\} $ in $\mathcal{X}_\tau $ such that each $x_ i$ is isomorphic to $f(u_ i)$ for some object $u_ i$ of $\mathcal{U}$,
the category $\mathcal{U}$ has equalizers, and
the functor $f$ is faithful.
Then there is a first quadrant spectral sequence of $\Gamma (\mathcal{O}_\mathcal {X})$-modules
\[ E_1^{p, q} = H^ q((\mathcal{U}_ p)_\tau , f_ p^*\mathcal{F}) \Rightarrow H^{p + q}(\mathcal{X}_\tau , \mathcal{F}) \]
converging to the cohomology of $\mathcal{F}$ in the $\tau $-topology.
Proof.
The proof of this lemma is identical to the proof of Lemma 96.19.8 except that it uses an injective resolution in $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$ and it uses Lemma 96.17.6 instead of Lemma 96.17.5.
$\square$
Here is a lemma that translates a more usual kind of covering in the kinds of coverings we have encountered above.
Lemma 96.19.10. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$.
Assume that $f$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Then for any object $y$ of $\mathcal{Y}$ there exists an fppf covering $\{ y_ i \to y\} $ and objects $x_ i$ of $\mathcal{X}$ such that $f(x_ i) \cong y_ i$ in $\mathcal{Y}$.
Assume that $f$ is representable by algebraic spaces, surjective, and smooth. Then for any object $y$ of $\mathcal{Y}$ there exists an étale covering $\{ y_ i \to y\} $ and objects $x_ i$ of $\mathcal{X}$ such that $f(x_ i) \cong y_ i$ in $\mathcal{Y}$.
Proof.
Proof of (1). Suppose that $y$ lies over the scheme $V$. We may think of $y$ as a morphism $(\mathit{Sch}/V)_{fppf} \to \mathcal{Y}$. By definition the $2$-fibre product $\mathcal{X} \times _\mathcal {Y} (\mathit{Sch}/V)_{fppf}$ is representable by an algebraic space $W$ and the morphism $W \to V$ is surjective, flat, and locally of finite presentation. Choose a scheme $U$ and a surjective étale morphism $U \to W$. Then $U \to V$ is also surjective, flat, and locally of finite presentation (see Morphisms of Spaces, Lemmas 67.39.7, 67.39.8, 67.5.4, 67.28.2, and 67.30.3). Hence $\{ U \to V\} $ is an fppf covering. Denote $x$ the object of $\mathcal{X}$ over $U$ corresponding to the $1$-morphism $(\mathit{Sch}/U)_{fppf} \to \mathcal{X}$. Then $\{ f(x) \to y\} $ is the desired fppf covering of $\mathcal{Y}$.
Proof of (2). Suppose that $y$ lies over the scheme $V$. We may think of $y$ as a morphism $(\mathit{Sch}/V)_{fppf} \to \mathcal{Y}$. By definition the $2$-fibre product $\mathcal{X} \times _\mathcal {Y} (\mathit{Sch}/V)_{fppf}$ is representable by an algebraic space $W$ and the morphism $W \to V$ is surjective and smooth. Choose a scheme $U$ and a surjective étale morphism $U \to W$. Then $U \to V$ is also surjective and smooth (see Morphisms of Spaces, Lemmas 67.39.6, 67.5.4, and 67.37.2). Hence $\{ U \to V\} $ is a smooth covering. By More on Morphisms, Lemma 37.38.7 there exists an étale covering $\{ V_ i \to V\} $ such that each $V_ i \to V$ factors through $U$. Denote $x_ i$ the object of $\mathcal{X}$ over $V_ i$ corresponding to the $1$-morphism
\[ (\mathit{Sch}/V_ i)_{fppf} \to (\mathit{Sch}/U)_{fppf} \to \mathcal{X}. \]
Then $\{ f(x_ i) \to y\} $ is the desired étale covering of $\mathcal{Y}$.
$\square$
Lemma 96.19.11. Let $f : \mathcal{U} \to \mathcal{X}$ and $g : \mathcal{X} \to \mathcal{Y}$ be composable $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, \linebreak[0] fppf\} $. Assume
$\mathcal{F}$ is an abelian sheaf on $\mathcal{X}_\tau $,
for every object $x$ of $\mathcal{X}$ there exists a covering $\{ x_ i \to x\} $ in $\mathcal{X}_\tau $ such that each $x_ i$ is isomorphic to $f(u_ i)$ for some object $u_ i$ of $\mathcal{U}$,
the category $\mathcal{U}$ has equalizers, and
the functor $f$ is faithful.
Then there is a first quadrant spectral sequence of abelian sheaves on $\mathcal{Y}_\tau $
\[ E_1^{p, q} = R^ q(g \circ f_ p)_*f_ p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F} \]
where all higher direct images are computed in the $\tau $-topology.
Proof.
Note that the assumptions on $f : \mathcal{U} \to \mathcal{X}$ and $\mathcal{F}$ are identical to those in Lemma 96.19.8. Hence the preliminary remarks made in the proof of that lemma hold here also. These remarks imply in particular that
\[ 0 \to g_*\mathcal{I} \to (g \circ f_0)_*f_0^{-1}\mathcal{I} \to (g \circ f_1)_*f_1^{-1}\mathcal{I} \to \ldots \]
is exact if $\mathcal{I}$ is an injective object of $\textit{Ab}(\mathcal{X}_\tau )$. Having said this, consider the two spectral sequences of Homology, Section 12.25 associated to the double complex $\mathcal{C}^{\bullet , \bullet }$ with terms
\[ \mathcal{C}^{p, q} = (g \circ f_ p)_*\mathcal{I}^ q \]
where $\mathcal{F} \to \mathcal{I}^\bullet $ is an injective resolution in $\textit{Ab}(\mathcal{X}_\tau )$. The first spectral sequence implies, via Homology, Lemma 12.25.4, that $g_*\mathcal{I}^\bullet $ is quasi-isomorphic to the total complex associated to $\mathcal{C}^{\bullet , \bullet }$. Since $f_ p^{-1}\mathcal{I}^\bullet $ is an injective resolution of $f_ p^{-1}\mathcal{F}$ (see Lemma 96.17.5) the second spectral sequence has terms $E_1^{p, q} = R^ q(g \circ f_ p)_*f_ p^{-1}\mathcal{F}$ as in the statement of the lemma.
$\square$
Lemma 96.19.12. Let $f : \mathcal{U} \to \mathcal{X}$ and $g : \mathcal{X} \to \mathcal{Y}$ be composable $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, \linebreak[0] fppf\} $. Assume
$\mathcal{F}$ is an object of $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$,
for every object $x$ of $\mathcal{X}$ there exists a covering $\{ x_ i \to x\} $ in $\mathcal{X}_\tau $ such that each $x_ i$ is isomorphic to $f(u_ i)$ for some object $u_ i$ of $\mathcal{U}$,
the category $\mathcal{U}$ has equalizers, and
the functor $f$ is faithful.
Then there is a first quadrant spectral sequence in $\textit{Mod}(\mathcal{Y}_\tau , \mathcal{O}_\mathcal {Y})$
\[ E_1^{p, q} = R^ q(g \circ f_ p)_*f_ p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F} \]
where all higher direct images are computed in the $\tau $-topology.
Proof.
The proof is identical to the proof of Lemma 96.19.11 except that it uses an injective resolution in $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$ and it uses Lemma 96.17.6 instead of Lemma 96.17.5.
$\square$
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Comment #1862 by Mao Li on
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