## 21.36 Derived lower shriek

In this section we study morphisms $g$ of ringed topoi where besides $Lg^*$ and $Rg_*$ there also exists a derived functor $Lg_!$.

Lemma 21.36.1. Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous and cocontinuous functor of sites. Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be the corresponding morphism of topoi. Let $\mathcal{O}_\mathcal {D}$ be a sheaf of rings and let $\mathcal{I}$ be an injective $\mathcal{O}_\mathcal {D}$-module. Then $H^ p(U, g^{-1}\mathcal{I}) = 0$ for all $p > 0$ and $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

Proof. The vanishing of the lemma follows from Lemma 21.10.9 if we can prove vanishing of all higher Čech cohomology groups $\check H^ p(\mathcal{U}, g^{-1}\mathcal{I})$ for any covering $\mathcal{U} = \{ U_ i \to U\}$ of $\mathcal{C}$. Since $u$ is continuous, $u(\mathcal{U}) = \{ u(U_ i) \to u(U)\}$ is a covering of $\mathcal{D}$, and $u(U_{i_0} \times _ U \ldots \times _ U U_{i_ n}) = u(U_{i_0}) \times _{u(U)} \ldots \times _{u(U)} u(U_{i_ n})$. Thus we have

$\check H^ p(\mathcal{U}, g^{-1}\mathcal{I}) = \check H^ p(u(\mathcal{U}), \mathcal{I})$

because $g^{-1} = u^ p$ by Sites, Lemma 7.21.5. Since $\mathcal{I}$ is an injective $\mathcal{O}_\mathcal {D}$-module these Čech cohomology groups vanish, see Lemma 21.12.3. $\square$

Lemma 21.36.2. Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous and cocontinuous functor of sites. Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be the corresponding morphism of topoi. Let $\mathcal{O}_\mathcal {D}$ be a sheaf of rings and set $\mathcal{O}_\mathcal {C} = g^{-1}\mathcal{O}_\mathcal {D}$. The functor $g_! : \textit{Mod}(\mathcal{O}_\mathcal {C}) \to \textit{Mod}(\mathcal{O}_\mathcal {D})$ (see Modules on Sites, Lemma 18.41.1) has a left derived functor

$Lg_! : D(\mathcal{O}_\mathcal {C}) \longrightarrow D(\mathcal{O}_\mathcal {D})$

which is left adjoint to $g^*$. Moreover, for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we have

$Lg_!(j_{U!}\mathcal{O}_ U) = g_!j_{U!}\mathcal{O}_ U = j_{u(U)!} \mathcal{O}_{u(U)}.$

where $j_{U!}$ and $j_{u(U)!}$ are extension by zero associated to the localization morphism $j_ U : \mathcal{C}/U \to \mathcal{C}$ and $j_{u(U)} : \mathcal{D}/u(U) \to \mathcal{D}$.

Proof. We are going to use Derived Categories, Proposition 13.29.2 to construct $Lg_!$. To do this we have to verify assumptions (1), (2), (3), (4), and (5) of that proposition. First, since $g_!$ is a left adjoint we see that it is right exact and commutes with all colimits, so (5) holds. Conditions (3) and (4) hold because the category of modules on a ringed site is a Grothendieck abelian category. Let $\mathcal{P} \subset \mathop{\mathrm{Ob}}\nolimits (\textit{Mod}(\mathcal{O}_\mathcal {C}))$ be the collection of $\mathcal{O}_\mathcal {C}$-modules which are direct sums of modules of the form $j_{U!}\mathcal{O}_ U$. Note that $g_!j_{U!}\mathcal{O}_ U = j_{u(U)!} \mathcal{O}_{u(U)}$, see proof of Modules on Sites, Lemma 18.41.1. Every $\mathcal{O}_\mathcal {C}$-module is a quotient of an object of $\mathcal{P}$, see Modules on Sites, Lemma 18.28.7. Thus (1) holds. Finally, we have to prove (2). Let $\mathcal{K}^\bullet$ be a bounded above acyclic complex of $\mathcal{O}_\mathcal {C}$-modules with $\mathcal{K}^ n \in \mathcal{P}$ for all $n$. We have to show that $g_!\mathcal{K}^\bullet$ is exact. To do this it suffices to show, for every injective $\mathcal{O}_\mathcal {D}$-module $\mathcal{I}$ that

$\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_\mathcal {D})}( g_!\mathcal{K}^\bullet , \mathcal{I}[n]) = 0$

for all $n \in \mathbf{Z}$. Since $\mathcal{I}$ is injective we have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_\mathcal {D})}( g_!\mathcal{K}^\bullet , \mathcal{I}[n]) & = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_\mathcal {D})}( g_!\mathcal{K}^\bullet , \mathcal{I}[n]) \\ & = H^ n(\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {D}}( g_!\mathcal{K}^\bullet , \mathcal{I})) \\ & = H^ n(\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {C}}( \mathcal{K}^\bullet , g^{-1}\mathcal{I})) \end{align*}

the last equality by the adjointness of $g_!$ and $g^{-1}$.

The vanishing of this group would be clear if $g^{-1}\mathcal{I}$ were an injective $\mathcal{O}_\mathcal {C}$-module. But $g^{-1}\mathcal{I}$ isn't necessarily an injective $\mathcal{O}_\mathcal {C}$-module as $g_!$ isn't exact in general. We do know that

$\mathop{\mathrm{Ext}}\nolimits ^ p_{\mathcal{O}_\mathcal {C}}( j_{U!}\mathcal{O}_ U, g^{-1}\mathcal{I}) = H^ p(U, g^{-1}\mathcal{I}) = 0 \text{ for }p \geq 1$

Here the first equality follows from $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {C}}(j_{U!}\mathcal{O}_ U, \mathcal{H}) = \mathcal{H}(U)$ and taking derived functors and the vanishing of $H^ p(U, g^{-1}\mathcal{I})$ for $p > 0$ and $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ follows from Lemma 21.36.1. Since each $\mathcal{K}^{-q}$ is a direct sum of modules of the form $j_{U!}\mathcal{O}_ U$ we see that

$\mathop{\mathrm{Ext}}\nolimits ^ p_{\mathcal{O}_\mathcal {C}}(\mathcal{K}^{-q}, g^{-1}\mathcal{I}) = 0 \text{ for }p \geq 1\text{ and all }q$

Let us use the spectral sequence (see Example 21.31.1)

$E_1^{p, q} = \mathop{\mathrm{Ext}}\nolimits ^ p_{\mathcal{O}_\mathcal {C}}( \mathcal{K}^{-q}, g^{-1}\mathcal{I}) \Rightarrow \mathop{\mathrm{Ext}}\nolimits ^{p + q}_{\mathcal{O}_\mathcal {C}}( \mathcal{K}^\bullet , g^{-1}\mathcal{I}) = 0.$

Note that the spectral sequence abuts to zero as $\mathcal{K}^\bullet$ is acyclic (hence vanishes in the derived category, hence produces vanishing ext groups). By the vanishing of higher exts proved above the only nonzero terms on the $E_1$ page are the terms $E_1^{0, q} = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {C}}( \mathcal{K}^{-q}, g^{-1}\mathcal{I})$. We conclude that the complex $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {C}}( \mathcal{K}^\bullet , g^{-1}\mathcal{I})$ is acyclic as desired.

Thus the left derived functor $Lg_!$ exists. It is left adjoint to $g^{-1} = g^* = Rg^* = Lg^*$, i.e., we have

21.36.2.1
$$\label{sites-cohomology-equation-to-prove} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_\mathcal {C})}(K, g^*L) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_\mathcal {D})}(Lg_!K, L)$$

by Derived Categories, Lemma 13.30.3. This finishes the proof. $\square$

Remark 21.36.3. Warning! Let $u : \mathcal{C} \to \mathcal{D}$, $g$, $\mathcal{O}_\mathcal {D}$, and $\mathcal{O}_\mathcal {C}$ be as in Lemma 21.36.2. In general it is not the case that the diagram

$\xymatrix{ D(\mathcal{O}_\mathcal {C}) \ar[r]_{Lg_!} \ar[d]_{forget} & D(\mathcal{O}_\mathcal {D}) \ar[d]^{forget} \\ D(\mathcal{C}) \ar[r]^{Lg^{Ab}_!} & D(\mathcal{D}) }$

commutes where the functor $Lg_!^{Ab}$ is the one constructed in Lemma 21.36.2 but using the constant sheaf $\mathbf{Z}$ as the structure sheaf on both $\mathcal{C}$ and $\mathcal{D}$. In general it isn't even the case that $g_! = g_!^{Ab}$ (see Modules on Sites, Remark 18.41.2), but this phenomenon can occur even if $g_! = g_!^{Ab}$! Namely, the construction of $Lg_!$ in the proof of Lemma 21.36.2 shows that $Lg_!$ agrees with $Lg_!^{\textit{Ab}}$ if and only if the canonical maps

$Lg^{Ab}_!j_{U!}\mathcal{O}_ U \longrightarrow j_{u(U)!}\mathcal{O}_{u(U)}$

are isomorphisms in $D(\mathcal{D})$ for all objects $U$ in $\mathcal{C}$. In general all we can say is that there exists a natural transformation

$Lg_!^{Ab} \circ forget \longrightarrow forget \circ Lg_!$

Lemma 21.36.4. Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous and cocontinuous functor of sites. Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be the corresponding morphism of topoi. Let $\mathcal{O}_\mathcal {D}$ be a sheaf of rings and let $\mathcal{I}$ be an injective $\mathcal{O}_\mathcal {D}$-module. If $g_!^{Sh} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ commutes with fibre products1, then $g^{-1}\mathcal{I}$ is totally acyclic.

Proof. We will use the criterion of Lemma 21.13.5. Condition (1) holds by Lemma 21.36.1. Let $K' \to K$ be a surjective map of sheaves of sets on $\mathcal{C}$. Since $g_!^{Sh}$ is a left adjoint, we see that $g_!^{Sh}K' \to g_!^{Sh}K$ is surjective. Observe that

\begin{align*} H^0(K' \times _ K \ldots \times _ K K', g^{-1}\mathcal{I}) & = H^0(g_!^{Sh}(K' \times _ K \ldots \times _ K K'), \mathcal{I}) \\ & = H^0(g_!^{Sh}K' \times _{g_!^{Sh}K} \ldots \times _{g_!^{Sh}K} g_!^{Sh}K', \mathcal{I}) \end{align*}

by our assumption on $g_!^{Sh}$. Since $\mathcal{I}$ is an injective module it is totally acyclic by Lemma 21.14.1 (applied to the identity). Hence we can use the converse of Lemma 21.13.5 to see that the complex

$0 \to H^0(K, g^{-1}\mathcal{I}) \to H^0(K', g^{-1}\mathcal{I}) \to H^0(K' \times _ K K', g^{-1}\mathcal{I}) \to \ldots$

is exact as desired. $\square$

Lemma 21.36.5. Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous and cocontinuous functor of sites. Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be the corresponding morphism of topoi. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

1. For $M$ in $D(\mathcal{D})$ we have $R\Gamma (U, g^{-1}M) = R\Gamma (u(U), M)$.

2. If $\mathcal{O}_\mathcal {D}$ is a sheaf of rings and $\mathcal{O}_\mathcal {C} = g^{-1}\mathcal{O}_\mathcal {D}$, then for $M$ in $D(\mathcal{O}_\mathcal {D})$ we have $R\Gamma (U, g^*M) = R\Gamma (u(U), M)$.

Proof. In the bounded below case (1) and (2) can be seen by representing $K$ by a bounded below complex of injectives and using Lemma 21.36.1 as well as Leray's acyclicity lemma. In the unbounded case, first note that (1) is a special case of (2). For (2) we can use

$R\Gamma (U, g^*M) = R\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {C}}(j_{U!}\mathcal{O}_ U, g^*M) = R\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {D}}(j_{u(U)!}\mathcal{O}_{u(U)}, M) = R\Gamma (u(U), M)$

where the middle equality is a consequence of Lemma 21.36.2. $\square$

Lemma 21.36.6. Assume given a commutative diagram

$\xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_{(g', (g')^\sharp )} \ar[d]_{(f', (f')^\sharp )} & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \ar[d]^{(f, f^\sharp )} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^{(g, g^\sharp )} & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) }$

of ringed topoi. Assume

1. $f$, $f'$, $g$, and $g'$ correspond to cocontinuous functors $u$, $u'$, $v$, and $v'$ as in Sites, Lemma 7.21.1,

2. $v \circ u' = u \circ v'$,

3. $v$ and $v'$ are continuous as well as cocontinuous,

4. for any object $V'$ of $\mathcal{D}'$ the functor ${}^{u'}_{V'}\mathcal{I} \to {}^{\ \ \ u}_{v(V')}\mathcal{I}$ given by $v$ is cofinal,

5. $g^{-1}\mathcal{O}_{\mathcal{D}} = \mathcal{O}_{\mathcal{D}'}$ and $(g')^{-1}\mathcal{O}_{\mathcal{C}} = \mathcal{O}_{\mathcal{C}'}$, and

6. $g'_! : \textit{Ab}(\mathcal{C}') \to \textit{Ab}(\mathcal{C})$ is exact2.

Then we have $Rf'_* \circ (g')^* = g^* \circ Rf_*$ as functors $D(\mathcal{O}_\mathcal {C}) \to D(\mathcal{O}_{\mathcal{D}'})$.

Proof. We have $g^* = Lg^* = g^{-1}$ and $(g')^* = L(g')^* = (g')^{-1}$ by condition (5). By Lemma 21.20.7 it suffices to prove the result on the derived category $D(\mathcal{C})$ of abelian sheaves. Choose an object $K \in D(\mathcal{C})$. Let $\mathcal{I}^\bullet$ be a K-injective complex of abelian sheaves on $\mathcal{C}$ representing $K$. By Derived Categories, Lemma 13.31.9 and assumption (6) we find that $(g')^{-1}\mathcal{I}^\bullet$ is a K-injective complex of abelian sheaves on $\mathcal{C}'$. By Modules on Sites, Lemma 18.41.3 we find that $f'_*(g')^{-1}\mathcal{I}^\bullet = g^{-1}f_*\mathcal{I}^\bullet$. Since $f_*\mathcal{I}^\bullet$ represents $Rf_*K$ and since $f'_*(g')^{-1}\mathcal{I}^\bullet$ represents $Rf'_*(g')^{-1}K$ we conclude. $\square$

Lemma 21.36.7. Consider a commutative diagram

$\xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}_{\mathcal{C}'} \ar[r]_{(g', (g')^\sharp )} \ar[d]_{(f', (f')^\sharp )} & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \ar[d]^{(f, f^\sharp )} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^{(g, g^\sharp )} & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) }$

of ringed topoi and suppose we have functors

$\xymatrix{ \mathcal{C}' \ar[r]_{v'} & \mathcal{C} \\ \mathcal{D}' \ar[r]^ v \ar[u]^{u'} & \mathcal{D} \ar[u]_ u }$

such that (with notation as in Sites, Sections 7.14 and 7.21) we have

1. $u$ and $u'$ are continuous and give rise to the morphisms $f$ and $f'$,

2. $v$ and $v'$ are cocontinuous giving rise to the morphisms $g$ and $g'$,

3. $u \circ v = v' \circ u'$,

4. $v$ and $v'$ are continuous as well as cocontinuous, and

5. $g^{-1}\mathcal{O}_{\mathcal{D}} = \mathcal{O}_{\mathcal{D}'}$ and $(g')^{-1}\mathcal{O}_{\mathcal{C}} = \mathcal{O}_{\mathcal{C}'}$.

Then $Rf'_* \circ (g')^* = g^* \circ Rf_*$ as functors $D^+(\mathcal{O}_\mathcal {C}) \to D^+(\mathcal{O}_{\mathcal{D}'})$. If in addition

1. $g'_! : \textit{Ab}(\mathcal{C}') \to \textit{Ab}(\mathcal{C})$ is exact3,

then $Rf'_* \circ (g')^* = g^* \circ Rf_*$ as functors $D(\mathcal{O}_\mathcal {C}) \to D(\mathcal{O}_{\mathcal{D}'})$.

Proof. We have $g^* = Lg^* = g^{-1}$ and $(g')^* = L(g')^* = (g')^{-1}$ by condition (5). By Lemma 21.20.7 it suffices to prove the result on the derived category $D^+(\mathcal{C})$ or $D(\mathcal{C})$ of abelian sheaves.

Choose an object $K \in D^+(\mathcal{C})$. Let $\mathcal{I}^\bullet$ be a bounded below complex of injective abelian sheaves on $\mathcal{C}$ representing $K$. By Lemma 21.36.1 we see that $H^ p(U', (g')^{-1}\mathcal{I}^ q) = 0$ for all $p > 0$ and any $q$ and any $U' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}')$. Recall that $R^ pf'_*(g')^{-1}\mathcal{I}^ q$ is the sheaf associated to the presheaf $V' \mapsto H^ p(u'(V'), (g')^{-1}\mathcal{I}^ q)$, see Lemma 21.7.4. Thus we see that $(g')^{-1}\mathcal{I}^ q$ is right acyclic for the functor $f'_*$. By Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) we find that $f'_*(g')^*\mathcal{I}^\bullet$ represents $Rf'_*(g')^{-1}K$. By Modules on Sites, Lemma 18.41.4 we find that $f'_*(g')^{-1}\mathcal{I}^\bullet = g^{-1}f_*\mathcal{I}^\bullet$. Since $g^{-1}f_*\mathcal{I}^\bullet$ represents $g^{-1}Rf_*K$ we conclude.

Choose an object $K \in D(\mathcal{C})$. Let $\mathcal{I}^\bullet$ be a K-injective complex of abelian sheaves on $\mathcal{C}$ representing $K$. By Derived Categories, Lemma 13.31.9 and assumption (6) we find that $(g')^{-1}\mathcal{I}^\bullet$ is a K-injective complex of abelian sheaves on $\mathcal{C}'$. By Modules on Sites, Lemma 18.41.4 we find that $f'_*(g')^{-1}\mathcal{I}^\bullet = g^{-1}f_*\mathcal{I}^\bullet$. Since $f_*\mathcal{I}^\bullet$ represents $Rf_*K$ and since $f'_*(g')^{-1}\mathcal{I}^\bullet$ represents $Rf'_*(g')^{-1}K$ we conclude. $\square$

[1] Holds if $\mathcal{C}$ has finite connected limits and $u$ commutes with them, see Sites, Lemma 7.21.6.
[2] Holds if fibre products and equalizers exist in $\mathcal{C}'$ and $v'$ commutes with them, see Modules on Sites, Lemma 18.16.3.
[3] Holds if fibre products and equalizers exist in $\mathcal{C}'$ and $v'$ commutes with them, see Modules on Sites, Lemma 18.16.3.

Comment #5481 by Zongzhu Lin on

In the first sentence of this section, "is" is missing. It should be "there isalso a derived functor Lg!."

Comment #5482 by Zongzhu Lin on

In the last sentence of proof of Lemma 0D6X, "since $\mathcal{I}$ is an injective $\mathcal{O}_{\mathcal{D}}$-module"

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