The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

21.36 Derived lower shriek

In this section we study morphisms $g$ of ringed topoi where besides $Lg^*$ and $Rg_*$ there also 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 21.11.9 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 $\mathcal{O}$-module these Čech cohomology groups vanish, see Lemma 21.13.3. $\square$

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 18.40.1) 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 13.28.2 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 18.40.1. 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 21.32.1)

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

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
\begin{equation} \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) \end{equation}

by Derived Categories, Lemma 13.28.5. 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 18.40.2), 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 limp.

Proof. We will use the criterion of Lemma 21.14.5. 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 injective module it is limp by Lemma 21.15.1 (applied to the identity). Hence we can use the converse of Lemma 21.14.5 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. $\square$

[1] Holds if $\mathcal{C}$ has finite connected limits and $u$ commutes with them, see Sites, Lemma 7.21.6.

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