36.6 Cohomology with support in a closed

We elaborate on the material in Cohomology, Sections 20.21 and 20.34 for schemes and quasi-coherent modules.

Definition 36.6.1. Let $X$ be a scheme. Let $E$ be an object of $D(\mathcal{O}_ X)$. Let $T \subset X$ be a closed subset. We say $E$ is supported on $T$ if the cohomology sheaves $H^ i(E)$ are supported on $T$.

We repeat some of the discussion from Cohomology, Section 20.34 in the situation of the definition. Let $X$ be a scheme. Let $T \subset X$ be a closed subset. The category of $\mathcal{O}_ X$-modules whose support is contained in $T$ is a Serre subcategory of the category of all $\mathcal{O}_ X$-modules, see Homology, Definition 12.10.1 and Modules, Lemma 17.5.2. In the following we will denote $D_ T(\mathcal{O}_ X)$ the strictly full, saturated triangulated subcategory of $D(\mathcal{O}_ X)$ consisting of objects supported on $T$, see Derived Categories, Section 13.17.

In the situation of Definition 36.6.1 denote $i : T \to X$ the inclusion map. Recall from Cohomology, Section 20.34 that in this situation we have a functor $R\mathcal{H}_ T : D(\mathcal{O}_ X) \to D(i^{-1}\mathcal{O}_ X)$ which is right adjoint to $i_* : D(i^{-1}\mathcal{O}_ X) \to D(\mathcal{O}_ X)$.

Lemma 36.6.2. Let $X$ be a scheme. Let $T \subset X$ be a closed subset such that $X \setminus T$ is a retrocompact open of $X$. Let $i : T \to X$ be the inclusion.

1. For $E$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ we have $i_*R\mathcal{H}_ T(E)$ in $D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$.

2. The functor $i_* \circ R\mathcal{H}_ T : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$ is right adjoint to the inclusion functor $D_{\mathit{QCoh}, T}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ X)$.

Proof. Set $U = X \setminus T$ and denote $j : U \to X$ the inclusion. By Cohomology, Lemma 20.34.6 there is a distinguished triangle

$i_*R\mathcal{H}_ T(E) \to E \to Rj_*(E|_ U) \to i_*R\mathcal{H}_ Z(E)$

in $D(\mathcal{O}_ X)$. By Lemma 36.4.1 the complex $Rj_*(E|_ U)$ has quasi-coherent cohomology sheaves (this is where we use that $U$ is retrocompact in $X$). Thus we see that (1) is true. Part (2) follows from this and the adjointness of functors in Cohomology, Lemma 20.34.2. $\square$

Lemma 36.6.3. Let $X$ be a scheme. Let $T \subset X$ be a closed subset such that $X \setminus T$ is a retrocompact open of $X$. Then for a family of objects $E_ i$, $i \in I$ of $D_\mathit{QCoh}(\mathcal{O}_ X)$ we have $R\mathcal{H}_ T(\bigoplus E_ i) = \bigoplus R\mathcal{H}_ T(E_ i)$.

Proof. Set $U = X \setminus T$ and denote $j : U \to X$ the inclusion. By Cohomology, Lemma 20.34.6 there is a distinguished triangle

$i_*R\mathcal{H}_ T(E) \to E \to Rj_*(E|_ U) \to i_*R\mathcal{H}_ Z(E)$

in $D(\mathcal{O}_ X)$ for any $E$ in $D(\mathcal{O}_ X)$. The functor $E \mapsto Rj_*(E|_ U)$ commutes with direct sums on $D_\mathit{QCoh}(\mathcal{O}_ X)$ by Lemma 36.4.5. It follows that the same is true for the functor $i_* \circ R\mathcal{H}_ T$ (details omitted). Since $i_* : D(i^{-1}\mathcal{O}_ X) \to D_ T(\mathcal{O}_ X)$ is an equivalence (Cohomology, Lemma 20.34.2) we conclude. $\square$

Remark 36.6.4. Let $X$ be a scheme. Let $f_1, \ldots , f_ c \in \Gamma (X, \mathcal{O}_ X)$. Denote $Z \subset X$ the closed subscheme cut out by $f_1, \ldots , f_ c$. For $0 \leq p < c$ and $1 \leq i_0 < \ldots < i_ p \leq c$ we denote $U_{i_0 \ldots i_ p} \subset X$ the open subscheme where $f_{i_0} \ldots f_{i_ p}$ is invertible. For any $\mathcal{O}_ X$-module $\mathcal{F}$ we set

$\mathcal{F}_{i_0 \ldots i_ p} = (U_{i_0 \ldots i_ p} \to X)_*(\mathcal{F}|_{U_{i_0 \ldots i_ p}})$

In this situation the extended alternating Čech complex is the complex of $\mathcal{O}_ X$-modules

36.6.4.1
\begin{equation} \label{perfect-equation-extended-alternating} 0 \to \mathcal{F} \to \bigoplus \nolimits _{i_0} \mathcal{F}_{i_0} \to \ldots \to \bigoplus \nolimits _{i_0 < \ldots < i_ p} \mathcal{F}_{i_0 \ldots i_ p} \to \ldots \to \mathcal{F}_{1 \ldots c} \to 0 \end{equation}

where $\mathcal{F}$ is put in degree $0$. The maps are constructed as follows. Given $1 \leq i_0 < \ldots < i_{p + 1} \leq c$ and $0 \leq j \leq p + 1$ we have the canonical map

$\mathcal{F}_{i_0 \ldots \hat i_ j \ldots i_{p + 1}} \to \mathcal{F}_{i_0 \ldots i_ p}$

coming from the inclusion $U_{i_0 \ldots i_ p} \subset U_{i_0 \ldots \hat i_ j \ldots i_{p + 1}}$. The differentials in the extended alternating complex use these canonical maps with sign $(-1)^ j$.

Lemma 36.6.5. With $X$, $f_1, \ldots , f_ c \in \Gamma (X, \mathcal{O}_ X)$, and $\mathcal{F}$ as in Remark 36.6.4 the complex (36.6.4.1) restricts to an acyclic complex over $X \setminus Z$.

We remark that this lemma holds more generally for any extended alternating Čech complex defined as in Remark 36.6.4 starting with a finite open covering $X \setminus Z = U_1 \cup \ldots \cup U_ c$.

Proof. Let $W \subset X \setminus Z$ be an open subset. Evaluating the complex of sheaves (36.6.4.1) on $W$ we obtain the complex

$\mathcal{F}(W) \to \bigoplus \nolimits _{i_0} \mathcal{F}(U_{i_0} \cap W) \to \bigoplus \nolimits _{i_0 < i_1} \mathcal{F}(U_{i_0i_1} \cap W) \to \ldots$

In other words, we obtain the extended ordered Čech complex for the covering $W = \bigcup U_ i \cap W$ and the standard ordering on $\{ 1, \ldots , c\}$, see Cohomology, Section 20.23. By Cohomology, Lemma 20.23.7 this complex is homotopic to zero as soon as $W$ is contained in $V(f_ i)$ for some $1 \leq i \leq c$. This finishes the proof. $\square$

Remark 36.6.6. Let $X$, $f_1, \ldots , f_ c \in \Gamma (X, \mathcal{O}_ X)$, and $\mathcal{F}$ be as in Remark 36.6.4. Denote $\mathcal{F}^\bullet$ the complex (36.6.4.1). By Lemma 36.6.5 the cohomology sheaves of $\mathcal{F}^\bullet$ are supported on $Z$ hence $\mathcal{F}^\bullet$ is an object of $D_ Z(\mathcal{O}_ X)$. On the other hand, the equality $\mathcal{F}^0 = \mathcal{F}$ determines a canonical map $\mathcal{F}^\bullet \to \mathcal{F}$ in $D(\mathcal{O}_ X)$. As $i_* \circ R\mathcal{H}_ Z$ is a right adjoint to the inclusion functor $D_ Z(\mathcal{O}_ X) \to D(\mathcal{O}_ X)$, see Cohomology, Lemma 20.34.2, we obtain a canonical commutative diagram

$\xymatrix{ \mathcal{F}^\bullet \ar[rd] \ar[rr] & & \mathcal{F} \\ & i_*R\mathcal{H}_ Z(\mathcal{F}) \ar[ru] }$

in $D(\mathcal{O}_ X)$ functorial in the $\mathcal{O}_ X$-module $\mathcal{F}$.

Lemma 36.6.7. With $X$, $f_1, \ldots , f_ c \in \Gamma (X, \mathcal{O}_ X)$, and $\mathcal{F}$ as in Remark 36.6.4. If $\mathcal{F}$ is quasi-coherent, then the complex (36.6.4.1) represents $i_* R\mathcal{H}_ Z(\mathcal{F})$ in $D_ Z(\mathcal{O}_ X)$.

Proof. Let us denote $\mathcal{F}^\bullet$ the complex (36.6.4.1). The statement of the lemma means that the map $\mathcal{F}^\bullet \to i_*R\mathcal{H}_ Z(\mathcal{F})$ of Remark 36.6.6 is an isomorphism. Since $\mathcal{F}^\bullet$ is in $D_ Z(\mathcal{O}_ X)$ (see remark cited), we see that $i_*R\mathcal{H}_ Z(\mathcal{F}^\bullet ) = \mathcal{F}^\bullet$ by Cohomology, Lemma 20.34.2. The morphism $U_{i_0 \ldots i_ p} \to X$ is affine as it is given over affine opens of $X$ by inverting the function $f_{i_0} \ldots f_{i_ p}$. Thus we see that

$\mathcal{F}_{i_0 \ldots i_ p} = (U_{i_0 \ldots i_ p} \to X)_*\mathcal{F}|_{U_{i_0 \ldots i_ p}} = R(U_{i_0 \ldots i_ p} \to X)_*\mathcal{F}|_{U_{i_0 \ldots i_ p}}$

by Cohomology of Schemes, Lemma 30.2.3 and the assumption that $\mathcal{F}$ is quasi-coherent. We conclude that $R\mathcal{H}_ Z(\mathcal{F}_{i_0 \ldots i_ p}) = 0$ by Cohomology, Lemma 20.34.7. Thus $i_*R\mathcal{H}_ Z(\mathcal{F}^ p) = 0$ for $p > 0$. Putting everything together we obtain

$\mathcal{F}^\bullet = i_*R\mathcal{H}_ Z(\mathcal{F}^\bullet ) = i_*R\mathcal{H}_ Z(\mathcal{F})$

as desired. $\square$

Lemma 36.6.8. Let $X$ be a scheme. Let $T \subset X$ be a closed subset which can locally be cut out by at most $c$ elements of the structure sheaf. Then $\mathcal{H}^ i_ Z(\mathcal{F}) = 0$ for $i > c$ and any quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$.

Proof. This follows immediately from the local description of $R\mathcal{H}_ T(\mathcal{F})$ given in Lemma 36.6.7. $\square$

Lemma 36.6.9. Let $X$ be a scheme. Let $T \subset X$ be a closed subset which can locally be cut out by a Koszul regular sequence having $c$ elements. Then $\mathcal{H}^ i_ Z(\mathcal{F}) = 0$ for $i \not= c$ for every flat, quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$.

Proof. By the description of $R\mathcal{H}_ Z(\mathcal{F})$ given in Lemma 36.6.7 this boils down to the following algebra statement: given a ring $R$, a Koszul regular sequence $f_1, \ldots , f_ c \in R$, and a flat $R$-module $M$, the extended alternating Čech complex $M \to \bigoplus \nolimits _{i_0} M_{f_{i_0}} \to \bigoplus \nolimits _{i_0 < i_1} M_{f_{i_0}f_{i_1}} \to \ldots \to M_{f_1 \ldots f_ c}$ from More on Algebra, Section 15.29 only has cohomology in degree $c$. By More on Algebra, Lemma 15.31.1 we obtain the desired vanishing for the extended alternating Čech complex of $R$. Since the complex for $M$ is obtained by tensoring this with the flat $R$-module $M$ (More on Algebra, Lemma 15.29.2) we conclude. $\square$

Remark 36.6.10. With $X$, $f_1, \ldots , f_ c \in \Gamma (X, \mathcal{O}_ X)$, and $\mathcal{F}$ as in Remark 36.6.4. There is a canonical $\mathcal{O}_ X|_ Z$-linear map

$c_{f_1, \ldots , f_ c} : i^*\mathcal{F} \longrightarrow \mathcal{H}^ c_ Z(\mathcal{F})$

functorial in $\mathcal{F}$. Namely, denoting $\mathcal{F}^\bullet$ the extended alternating Čech complex (36.6.4.1) we have the canonical map $\mathcal{F}^\bullet \to i_*R\mathcal{H}_ Z(\mathcal{F})$ of Remark 36.6.6. This determines a canonical map

$\mathop{\mathrm{Coker}}\left(\bigoplus \mathcal{F}_{1 \ldots \hat i \ldots c} \to \mathcal{F}_{1 \ldots c}\right) \longrightarrow i_*\mathcal{H}^ c_ Z(\mathcal{F})$

on cohomology sheaves in degree $c$. Given a local section $s$ of $\mathcal{F}$ we can consider the local section

$\frac{s}{f_1 \ldots f_ c}$

of $\mathcal{F}_{1 \ldots c}$. The class of this section in the cokernel displayed above depends only on $s$ modulo the image of $(f_1, \ldots , f_ c) : \mathcal{F}^{\oplus c} \to \mathcal{F}$. Since $i_*i^*\mathcal{F}$ is equal to the cokernel of $(f_1, \ldots , f_ c) : \mathcal{F}^{\oplus c} \to \mathcal{F}$ we see that we get an $\mathcal{O}_ X$-module map $i_*i^*\mathcal{F} \to i_*\mathcal{H}_ Z^ c(\mathcal{F})$. As $i_*$ is fully faithful we get the map $c_{f_1, \ldots , f_ c}$.

Example 36.6.11. Let $X = \mathop{\mathrm{Spec}}(A)$ be affine, $f_1, \ldots , f_ c \in A$, and let $\mathcal{F} = \widetilde{M}$ for some $A$-module $M$. The map $c_{f_1, \ldots , f_ c}$ of Remark 36.6.10 can be described as the map

$M/(f_1, \ldots , f_ c)M \longrightarrow \mathop{\mathrm{Coker}}\left( \bigoplus M_{f_1 \ldots \hat f_ i \ldots f_ c} \to M_{f_1 \ldots f_ c} \right)$

sending the class of $s \in M$ to the class of $s/f_1 \ldots f_ c$ in the cokernel.

Lemma 36.6.12. With $X$, $f_1, \ldots , f_ c \in \Gamma (X, \mathcal{O}_ X)$, and $\mathcal{F}$ as in Remark 36.6.4. Let $a_{ji} \in \Gamma (X, \mathcal{O}_ X)$ for $1 \leq i, j \leq c$ and set $g_ j = \sum _{i = 1, \ldots , c} a_{ji}f_ i$. Assume $g_1, \ldots , g_ c$ scheme theoretically cut out $Z$. If $\mathcal{F}$ is quasi-coherent, then

$c_{f_1, \ldots , f_ c} = \det (a_{ji}) c_{g_1, \ldots , g_ c}$

where $c_{f_1, \ldots , f_ c}$ and $c_{g_1, \ldots , g_ c}$ are as in Remark 36.6.10.

Proof. We will prove that $c_{f_1, \ldots , f_ c}(s) = \det (a_{ij}) c_{g_1, \ldots , g_ c}(s)$ as global sections of $\mathcal{H}_ Z(\mathcal{F})$ for any $s \in \mathcal{F}(X)$. This is sufficient since we then obtain the same result for section over any open subscheme of $X$. To do this, for $1 \leq i_0 < \ldots < i_ p \leq c$ and $1 \leq j_0 < \ldots < j_ q \leq c$ we denote $U_{i_0 \ldots i_ p} \subset X$, $V_{j_0 \ldots j_ q} \subset X$, and $W_{i_0 \ldots i_ p, j_0 \ldots j_ q} \subset X$ the open subscheme where $f_{i_0} \ldots f_{i_ p}$ is invertible, $g_{j_0} \ldots g_{j_ q}$ is invertible, and where $f_{i_0} \ldots f_{i_ p}g_{j_0} \ldots g_{j_ q}$ is invertible. We denote $\mathcal{F}_{i_0 \ldots i_ p}$, resp. $\mathcal{F}'_{j_0 \ldots j_ q}$ $\mathcal{F}''_{i_0 \ldots i_ p, j_0 \ldots j_ q}$ the pushforward to $X$ of the restriction of $\mathcal{F}$ to $U_{i_0 \ldots i_ p}$, resp. $V_{j_0 \ldots j_ q}$, resp. $W_{i_0 \ldots i_ p, j_0 \ldots j_ q}$. Then we obtain three extended alternating Čech complexes

$\mathcal{F}^\bullet : \mathcal{F} \to \bigoplus \nolimits _{i_0} \mathcal{F}_{i_0} \to \bigoplus \nolimits _{i_0 < i_1} \mathcal{F}_{i_0i_1} \to \ldots$

and

$(\mathcal{F}')^\bullet : \mathcal{F} \to \bigoplus \nolimits _{j_0} \mathcal{F}'_{j_0} \to \bigoplus \nolimits _{j_0 < j_1} \mathcal{F}'_{j_0j_1} \to \ldots$

and

$(\mathcal{F}'')^\bullet : \mathcal{F} \to \bigoplus \nolimits _{i_0} \mathcal{F}_{i_0} \oplus \bigoplus \nolimits _{j_0} \mathcal{F}'_{j_0} \to \bigoplus \nolimits _{i_0 < i_1} \mathcal{F}_{i_0i_1} \oplus \bigoplus \nolimits _{i_0, j_0} \mathcal{F}''_{i_0, j_0} \oplus \bigoplus \nolimits _{j_0 < j_1} \mathcal{F}'_{j_0j_1} \to \ldots$

whose differentials are those used in defining (36.6.4.1). There are maps of complexes

$(\mathcal{F}'')^\bullet \to \mathcal{F}^\bullet \quad \text{and}\quad (\mathcal{F}'')^\bullet \to (\mathcal{F}')^\bullet$

given by the projection maps on the terms (and hence inducing the identity map in degree $0$). Observe that by Lemma 36.6.7 each of these complexes represents $i_*R\mathcal{H}_ Z(\mathcal{F})$ and these maps represent the identity on this object. Thus it suffices to find an element

$\sigma \in H^ c((\mathcal{F}'')^\bullet (X))$

mapping to $c_{f_1, \ldots , f_ c}(s)$ and $\det (a_{ji})c_{g_1, \ldots , g_ c}(s)$ by these two maps. It turns out we can explicitly give a cocycle for $\sigma$. Namely, we take

$\sigma _{1 \ldots c} = \frac{s}{f_1 \ldots f_ c} \in \mathcal{F}_{1 \ldots c}(X) \quad \text{and}\quad \sigma '_{1 \ldots c} = \frac{\det (a_{ji})s}{g_1 \ldots g_ c} \in \mathcal{F}'_{1 \ldots c}(X)$

and we take

$\sigma _{i_0 \ldots i_ p, j_0 \ldots j_{c - p - 2}} = \frac{\lambda (i_0 \ldots i_ p, j_0 \ldots j_{c - p - 2})s}{f_{i_0} \ldots f_{i_ p}g_{j_0} \ldots g_{j_{c - p - 2}}} \in \mathcal{F}''_{i_0 \ldots i_ p, j_0 \ldots j_{c - p - 2}}(X)$

where $\lambda (i_0 \ldots i_ p, j_0 \ldots j_{c - p - 2})$ is the coefficient of $e_1 \wedge \ldots \wedge e_ c$ in the formal expresssion

$e_{i_0} \wedge \ldots \wedge e_{i_ p} \wedge (a_{j_01} e_1 + \ldots + a_{j_0c}e_ c) \wedge \ldots \wedge (a_{j_{c - p - 2}1} e_1 + \ldots + a_{j_{c - p - 2}c}e_ c)$

To verify that $\sigma$ is a cocycle, we have to show for $1 \leq i_0 < \ldots < i_ p \leq c$ and $1 \leq j_0 < \ldots < j_{c - p - 1} \leq c$ that we have

\begin{align*} 0 & = \sum \nolimits _{a = 0, \ldots , p} (-1)^ a f_{i_ a} \lambda (i_0 \ldots \hat i_ a \ldots i_ p, j_0 \ldots j_{c - p - 1}) \\ & + \sum \nolimits _{b = 0, \ldots , c - p - 1} (-1)^{p + b + 1}g_{j_ b} \lambda (i_0 \ldots i_ p, j_0 \ldots \hat j_ b \ldots j_{c - p - 1}) \end{align*}

The easiest way to see this is perhaps to argue that the formal expression

$\xi = e_{i_0} \wedge \ldots \wedge e_{i_ p} \wedge (a_{j_01} e_1 + \ldots + a_{j_0c}e_ c) \wedge \ldots \wedge (a_{j_{c - p - 1}1} e_1 + \ldots + a_{j_{c - p - 1}c}e_ c)$

is $0$ as it is an element of the $(c + 1)$st wedge power of the free module on $e_1, \ldots , e_ c$ and that the expression above is the image of $\xi$ under the Koszul differential sending $e_ i \to f_ i$. Some details omitted. $\square$

Lemma 36.6.13. Let $X$ be a scheme. Let $Z \to X$ be a closed immersion of finite presentation whose conormal sheaf $\mathcal{C}_{Z/X}$ is locally free of rank $c$. Then there is a canonical map

$c : \wedge ^ c(\mathcal{C}_{Z/X})^\vee \otimes _{\mathcal{O}_ Z} i^*\mathcal{F} \longrightarrow \mathcal{H}_ Z^ c(\mathcal{F})$

functorial in the quasi-coherent module $\mathcal{F}$.

Proof. Follows from the construction in Remark 36.6.10 and the independence of the choice of generators of the ideal sheaf shown in Lemma 36.6.12. Some details omitted. $\square$

Remark 36.6.14. Let $g : X' \to X$ be a morphism of schemes. Let $f_1, \ldots , f_ c \in \Gamma (X, \mathcal{O}_ X)$. Set $f'_ i = g^\sharp (f_ i) \in \Gamma (X', \mathcal{O}_{X'})$. Denote $Z \subset X$, resp. $Z' \subset X'$ the closed subscheme cut out by $f_1, \ldots , f_ c$, resp. $f'_1, \ldots , f'_ c$. Then $Z' = Z \times _ X X'$. Denote $h : Z' \to Z$ the induced morphism of schemes. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. Set $\mathcal{F}' = g^*\mathcal{F}$. In this setting, if $\mathcal{F}$ is quasi-coherent, then the diagram

$\xymatrix{ (i')^{-1}\mathcal{O}_{X'} \otimes _{h^{-1}i^{-1}\mathcal{O}_ X} h^{-1}\mathcal{H}^ c_ Z(\mathcal{F}) \ar[r] & \mathcal{H}_{Z'}^ c(\mathcal{F}') \\ h^*i^*\mathcal{F} \ar[r] \ar[u]_-{c_{f_1, \ldots , f_ c}} & (i')^*\mathcal{F}' \ar[u]^-{c_{f'_1, \ldots , f'_ c}} }$

is commutative where the top horizonal arrow is the map of Cohomology, Remark 20.34.12 on cohomology sheaves in degree $c$. Namely, denote $\mathcal{F}^\bullet$, resp. $(\mathcal{F}')^\bullet$ the extended alternating Čech complex constructed in Remark 36.6.4 using $\mathcal{F}, f_1, \ldots , f_ c$, resp. $\mathcal{F}', f'_1, \ldots , f'_ c$. Note that $(\mathcal{F}')^\bullet = g^*\mathcal{F}^\bullet$. Then, without assuming $\mathcal{F}$ is quasi-coherent, the diagram

$\xymatrix{ i'_* L(g|_{Z'})^* R\mathcal{H}_ Z(\mathcal{F}) \ar[r] \ar@{=}[d] & i'_*R\mathcal{H}_{Z'}(Lg^*\mathcal{F}) \ar[d] \\ Lg^*i_*R\mathcal{H}_ Z(\mathcal{F}) & i'_*R\mathcal{H}_{Z'}(\mathcal{F}') \\ Lg^*(\mathcal{F}^\bullet ) \ar[u] \ar[r] & (\mathcal{F}')^\bullet \ar[u] }$

is commutative where $g|_{Z'} : (Z', (i')^{-1}\mathcal{O}_{X'}) \to (Z, i^{-1}\mathcal{O}_ X)$ is the induced morphism of ringed spaces. Here the top horizontal arrow is given in Cohomology, Remark 20.34.12 as is the explanation for the equal sign. The arrows pointing up are from Remark 36.6.6. The lower horizonal arrow is the map $Lg^*\mathcal{F}^\bullet \to g^*\mathcal{F}^\bullet = (\mathcal{F}')^\bullet$ and the arrow pointing down is induced by $Lg^*\mathcal{F} \to g^*\mathcal{F} = \mathcal{F}'$. The diagram commutes because going around the diagram both ways we obtain two arrows $Lg^*\mathcal{F}^\bullet \to i'_*R\mathcal{H}_{Z'}(\mathcal{F}')$ whose composition with $i'_*R\mathcal{H}_{Z'}(\mathcal{F}') \to \mathcal{F}'$ is the canonical map $Lg^*\mathcal{F}^\bullet \to \mathcal{F}'$. Some details omitted. Now the commutativity of the first diagram follows by looking at this diagram on cohomology sheaves in degree $c$ and using that the construction of the map $i^*\mathcal{F} \to \mathop{\mathrm{Coker}}(\bigoplus \mathcal{F}_{1 \ldots \hat i \ldots c} \to \mathcal{F}_{1 \ldots c})$ used in Remark 36.6.10 is compatible with pullbacks.

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