## 95.19 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 (95.18.0.1). 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.

Remark 95.19.1. We can define the complex $\mathcal{K}^\bullet (f, \mathcal{F})$ also if $\mathcal{F}$ is a presheaf, only we cannot use the reference to Sites, Section 7.45 to define the pullback maps. To explain the pullback maps, suppose given a commutative diagram

$\xymatrix{ \mathcal{V} \ar[rd]_ g \ar[rr]_ h & & \mathcal{U} \ar[ld]^ f \\ & \mathcal{X} }$

of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ and a presheaf $\mathcal{G}$ on $\mathcal{U}$ we can define the pullback map $f_*\mathcal{G} \to g_*h^{-1}\mathcal{G}$ as the composition

$f_*\mathcal{G} \longrightarrow f_*h_*h^{-1}\mathcal{G} = g_*h^{-1}\mathcal{G}$

where the map comes from the adjunction map $\mathcal{G} \to h_*h^{-1}\mathcal{G}$. This works because in our situation the functors $h_*$ and $h^{-1}$ are adjoint in presheaves (and agree with their counter parts on sheaves). See Sections 95.3 and 95.4.

Lemma 95.19.2. Generalities on relative Čech complexes.

1. 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 $e^{-1}\mathcal{K}^\bullet (f, \mathcal{F}) \to \mathcal{K}^\bullet (g, e^{-1}\mathcal{F})$.

2. if $h$ and $e$ are equivalences, then the map of (1) is an isomorphism,

3. 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 95.18.1 using the pullback maps of Remark 95.19.1. $\square$

Lemma 95.19.3. 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 95.18.2. $\square$

Remark 95.19.4. Let us “compute” the value of the relative Čech complex on an object $x$ of $\mathcal{X}$. Say $p(x) = U$. Consider the $2$-fibre product diagram (which serves to introduce the notation $g : \mathcal{V} \to \mathcal{Y}$)

$\xymatrix{ \mathcal{V} \ar@{=}[r] \ar[d]_ g & (\mathit{Sch}/U)_{fppf} \times _{x, \mathcal{X}} \mathcal{U} \ar[r] \ar[d] & \mathcal{U} \ar[d]^ f \\ \mathcal{Y} \ar@{=}[r] & (\mathit{Sch}/U)_{fppf} \ar[r]^-x & \mathcal{X} }$

Note that the morphism $\mathcal{V}_ n \to \mathcal{U}_ n$ of the proof of Lemma 95.18.1 induces an equivalence $\mathcal{V}_ n = (\mathit{Sch}/U)_{fppf} \times _{x, \mathcal{X}} \mathcal{U}_ n$. Hence we see from (95.5.0.1) that

$\Gamma (x, \mathcal{K}^\bullet (f, \mathcal{F})) = \check{\mathcal{C}}^\bullet (\mathcal{V} \to \mathcal{Y}, x^{-1}\mathcal{F})$

In words: The value of the relative Čech complex on an object $x$ of $\mathcal{X}$ is the Čech complex of the base change of $f$ to $\mathcal{X}/x \cong (\mathit{Sch}/U)_{fppf}$. This implies for example that Lemma 95.18.2 implies Lemma 95.19.3 and more generally that results on the (usual) Čech complex imply results for the relative Čech complex.

$\xymatrix{ \mathcal{V} \ar[d]_ g \ar[r]_ h & \mathcal{U} \ar[d]^ f \\ \mathcal{Y} \ar[r]^ e & \mathcal{X} }$

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 95.19.2 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 95.19.4. 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 95.19.4. 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 95.18.1. $\square$

Exactness can be checked on a “covering”.

Lemma 95.19.6. 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

1. 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

2. $f^{-1}\mathcal{F} \to f^{-1}\mathcal{G} \to f^{-1}\mathcal{H}$ 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 95.19.7. 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

1. $\mathcal{F}$ is an abelian sheaf on $\mathcal{X}_\tau$, and

2. 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 95.19.6 it suffices to check exactness after pulling back to $\mathcal{U}$. By Lemma 95.19.5 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 95.19.3. $\square$

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 95.19.8. 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

1. $\mathcal{F}$ is an abelian sheaf on $\mathcal{X}_\tau$,

2. 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 $\mathcal{U}$,

3. the category $\mathcal{U}$ has equalizers, and

4. 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 95.17.4 (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 95.17.5 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 95.17.1. Hence Proposition 95.19.7 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 95.19.9. 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

1. $\mathcal{F}$ is an object of $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$,

2. 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 $\mathcal{U}$,

3. the category $\mathcal{U}$ has equalizers, and

4. 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 95.19.8 except that it uses an injective resolution in $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$ and it uses Lemma 95.17.6 instead of Lemma 95.17.5. $\square$

Here is a lemma that translates a more usual kind of covering in the kinds of coverings we have encountered above.

Lemma 95.19.10. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$.

1. 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 $\mathcal{Y}$.

2. 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 66.39.7, 66.39.8, 66.5.4, 66.28.2, and 66.30.3). 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 66.39.6, 66.5.4, and 66.37.2). 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 95.19.11. 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

1. $\mathcal{F}$ is an abelian sheaf on $\mathcal{X}_\tau$,

2. 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 $\mathcal{U}$,

3. the category $\mathcal{U}$ has equalizers, and

4. 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 95.19.8. 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 95.17.5) 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 95.19.12. 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

1. $\mathcal{F}$ is an object of $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$,

2. 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 $\mathcal{U}$,

3. the category $\mathcal{U}$ has equalizers, and

4. 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 95.19.11 except that it uses an injective resolution in $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$ and it uses Lemma 95.17.6 instead of Lemma 95.17.5. $\square$

Comment #1862 by Mao Li on

The spectral sequences in this section should start in $E_{1}^{p,q}$ instead of $E_{2}^{p,q}$

Comment #1899 by on

Oops! Very bad with numbering spectral sequences (basically because I personally don't think the numbering should be standardized). Yes, from the construction of the spectral sequence in Section 12.25 it hould be $E_1$ and not $E_2$. Thanks! Corresponding changes can be found here.

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