21.38 Derived lower shriek for fibred categories
In this section we work out some special cases of the situation discussed in Section 21.37. We make sure that we have equality between lower shriek on modules and sheaves of abelian groups. We encourage the reader to skip this section on a first reading.
Situation 21.38.1. Here $(\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a ringed site and $p : \mathcal{C} \to \mathcal{D}$ is a fibred category. We endow $\mathcal{C}$ with the topology inherited from $\mathcal{D}$ (Stacks, Section 8.10). We denote $\pi : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ the morphism of topoi associated to $p$ (Stacks, Lemma 8.10.3). We set $\mathcal{O}_\mathcal {C} = \pi ^{-1}\mathcal{O}_\mathcal {D}$ so that we obtain a morphism of ringed topoi
\[ \pi : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) \]
Lemma 21.38.2. Assumptions and notation as in Situation 21.38.1. For $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ consider the induced morphism of topoi
\[ \pi _ U : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/p(U)) \]
Then there exists a morphism of topoi
\[ \sigma : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/p(U)) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \]
such that $\pi _ U \circ \sigma = \text{id}$ and $\sigma ^{-1} = \pi _{U, *}$.
Proof.
Observe that $\pi _ U$ is the restriction of $\pi $ to the localizations, see Sites, Lemma 7.28.4. For an object $V \to p(U)$ of $\mathcal{D}/p(U)$ denote $V \times _{p(U)} U \to U$ the strongly cartesian morphism of $\mathcal{C}$ over $\mathcal{D}$ which exists as $p$ is a fibred category. The functor
\[ v : \mathcal{D}/p(U) \to \mathcal{C}/U,\quad V/p(U) \mapsto V \times _{p(U)} U/U \]
is continuous by the definition of the topology on $\mathcal{C}$. Moreover, it is a right adjoint to $p$ by the definition of strongly cartesian morphisms. Hence we are in the situation discussed in Sites, Section 7.22 and we see that the sheaf $\pi _{U, *}\mathcal{F}$ is equal to $V \mapsto \mathcal{F}(V \times _{p(U)} U)$ (see especially Sites, Lemma 7.22.2).
But here we have more. Namely, the functor $v$ is also cocontinuous (as all morphisms in coverings of $\mathcal{C}$ are strongly cartesian). Hence $v$ defines a morphism $\sigma $ as indicated in the lemma. The equality $\sigma ^{-1} = \pi _{U, *}$ is immediate from the definition. Since $\pi _ U^{-1}\mathcal{G}$ is given by the rule $U'/U \mapsto \mathcal{G}(p(U')/p(U))$ it follows that $\sigma ^{-1} \circ \pi _ U^{-1} = \text{id}$ which proves the equality $\pi _ U \circ \sigma = \text{id}$.
$\square$
Situation 21.38.3. Let $(\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a ringed site. Let $u : \mathcal{C}' \to \mathcal{C}$ be a $1$-morphism of fibred categories over $\mathcal{D}$ (Categories, Definition 4.33.9). Endow $\mathcal{C}$ and $\mathcal{C}'$ with their inherited topologies (Stacks, Definition 8.10.2) and let $\pi : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$, $\pi ' : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$, and $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be the corresponding morphisms of topoi (Stacks, Lemma 8.10.3). Set $\mathcal{O}_\mathcal {C} = \pi ^{-1}\mathcal{O}_\mathcal {D}$ and $\mathcal{O}_{\mathcal{C}'} = (\pi ')^{-1}\mathcal{O}_\mathcal {D}$. Observe that $g^{-1}\mathcal{O}_\mathcal {C} = \mathcal{O}_{\mathcal{C}'}$ so that
\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[rd]_{\pi '} \ar[rr]_ g & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \ar[ld]^\pi \\ & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) } \]
is a commutative diagram of morphisms of ringed topoi.
Lemma 21.38.4. Assumptions and notation as in Situation 21.38.3. For $U' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}')$ set $U = u(U')$ and $V = p'(U')$ and consider the induced morphisms of ringed topoi
\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'/U'), \mathcal{O}_{U'}) \ar[rd]_{\pi '_{U'}} \ar[rr]_{g'} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_ U) \ar[ld]^{\pi _ U} \\ & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V), \mathcal{O}_ V) } \]
Then there exists a morphism of topoi
\[ \sigma ' : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}'/U'), \]
such that setting $\sigma = g' \circ \sigma '$ we have $\pi '_{U'} \circ \sigma ' = \text{id}$, $\pi _ U \circ \sigma = \text{id}$, $(\sigma ')^{-1} = \pi '_{U', *}$, and $\sigma ^{-1} = \pi _{U, *}$.
Proof.
Let $v' : \mathcal{D}/V \to \mathcal{C}'/U'$ be the functor constructed in the proof of Lemma 21.38.2 starting with $p' : \mathcal{C}' \to \mathcal{D}'$ and the object $U'$. Since $u$ is a $1$-morphism of fibred categories over $\mathcal{D}$ it transforms strongly cartesian morphisms into strongly cartesian morphisms, hence the functor $v = u \circ v'$ is the functor of the proof of Lemma 21.38.2 relative to $p : \mathcal{C} \to \mathcal{D}$ and $U$. Thus our lemma follows from that lemma.
$\square$
Lemma 21.38.5. Assumption and notation as in Situation 21.38.3.
There are left adjoints $g_! : \textit{Mod}(\mathcal{O}_{\mathcal{C}'}) \to \textit{Mod}(\mathcal{O}_\mathcal {C})$ and $g_!^{\textit{Ab}} : \textit{Ab}(\mathcal{C}') \to \textit{Ab}(\mathcal{C})$ to $g^* = g^{-1}$ on modules and on abelian sheaves.
The diagram
\[ \xymatrix{ \textit{Mod}(\mathcal{O}_{\mathcal{C}'}) \ar[d] \ar[r]_{g_!} & \textit{Mod}(\mathcal{O}_\mathcal {C}) \ar[d] \\ \textit{Ab}(\mathcal{C}') \ar[r]^{g_!^{\textit{Ab}}} & \textit{Ab}(\mathcal{C}) } \]
commutes.
There are left adjoints $Lg_! : D(\mathcal{O}_{\mathcal{C}'}) \to D(\mathcal{O}_\mathcal {C})$ and $Lg_!^{\textit{Ab}} : D(\mathcal{C}') \to D(\mathcal{C})$ to $g^* = g^{-1}$ on derived categories of modules and abelian sheaves.
The diagram
\[ \xymatrix{ D(\mathcal{O}_{\mathcal{C}'}) \ar[d] \ar[r]_{Lg_!} & D(\mathcal{O}_\mathcal {C}) \ar[d] \\ D(\mathcal{C}') \ar[r]^{Lg_!^{\textit{Ab}}} & D(\mathcal{C}) } \]
commutes.
Proof.
The functor $u$ is continuous and cocontinuous Stacks, Lemma 8.10.3. Hence the existence of the functors $g_!$, $g_!^{\textit{Ab}}$, $Lg_!$, and $Lg_!^{\textit{Ab}}$ can be found in Modules on Sites, Sections 18.16 and 18.41 and Section 21.37.
To prove (2) it suffices to show that the canonical map
\[ g_!^{\textit{Ab}}j_{U'!}\mathcal{O}_{U'} \to j_{u(U')!}\mathcal{O}_{u(U')} \]
is an isomorphism for all objects $U'$ of $\mathcal{C}'$, see Modules on Sites, Remark 18.41.2. Similarly, to prove (4) it suffices to show that the canonical map
\[ Lg_!^{\textit{Ab}}j_{U'!}\mathcal{O}_{U'} \to j_{u(U')!}\mathcal{O}_{u(U')} \]
is an isomorphism in $D(\mathcal{C})$ for all objects $U'$ of $\mathcal{C}'$, see Remark 21.37.3. This will also imply the previous formula hence this is what we will show.
We will use that for a localization morphism $j$ the functors $j_!$ and $j_!^{\textit{Ab}}$ agree (see Modules on Sites, Remark 18.19.6) and that $j_!$ is exact (Modules on Sites, Lemma 18.19.3). Let us adopt the notation of Lemma 21.38.4. Since $Lg_!^{\textit{Ab}} \circ j_{U'!} = j_{U!} \circ L(g')^{\textit{Ab}}_!$ (by commutativity of Sites, Lemma 7.28.4 and uniqueness of adjoint functors) it suffices to prove that $L(g')^{\textit{Ab}}_!\mathcal{O}_{U'} = \mathcal{O}_ U$. Using the results of Lemma 21.38.4 we have for any object $E$ of $D(\mathcal{C}/u(U'))$ the following sequence of equalities
\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{C}/U)}(L(g')_!^{\textit{Ab}}\mathcal{O}_{U'}, E) & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{C}'/U')}(\mathcal{O}_{U'}, (g')^{-1}E) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{C}'/U')}((\pi '_{U'})^{-1}\mathcal{O}_ V, (g')^{-1}E) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{D}/V)}(\mathcal{O}_ V, R\pi '_{U', *}(g')^{-1}E) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{D}/V)}(\mathcal{O}_ V, (\sigma ')^{-1}(g')^{-1}E) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{D}/V)}(\mathcal{O}_ V, \sigma ^{-1}E) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{D}/V)}(\mathcal{O}_ V, \pi _{U, *}E) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{C}/U)}(\pi _ U^{-1}\mathcal{O}_ V, E) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{C}/U)}(\mathcal{O}_ U, E) \end{align*}
By Yoneda's lemma we conclude.
$\square$
Lemma 21.38.8. Assumptions and notation as in Situation 21.38.1. For $\mathcal{F}$ in $\textit{Ab}(\mathcal{C})$ the sheaf $\pi _!\mathcal{F}$ is the sheaf associated to the presheaf
\[ V \longmapsto \mathop{\mathrm{colim}}\nolimits _{\mathcal{C}_ V^{opp}} \mathcal{F}|_{\mathcal{C}_ V} \]
with restriction maps as indicated in the proof.
Proof.
Denote $\mathcal{H}$ be the rule of the lemma. For a morphism $h : V' \to V$ of $\mathcal{D}$ there is a pullback functor $h^* : \mathcal{C}_ V \to \mathcal{C}_{V'}$ of fibre categories (Categories, Definition 4.33.6). Moreover for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ V)$ there is a strongly cartesian morphism $h^*U \to U$ covering $h$. Restriction along these strongly cartesian morphisms defines a transformation of functors
\[ \mathcal{F}|_{\mathcal{C}_ V} \longrightarrow \mathcal{F}|_{\mathcal{C}_{V'}} \circ h^*. \]
Hence a map $\mathcal{H}(V) \to \mathcal{H}(V')$ between colimits, see Categories, Lemma 4.14.8.
To prove the lemma we show that
\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{D})}(\mathcal{H}, \mathcal{G}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{F}, \pi ^{-1}\mathcal{G}) \]
for every sheaf $\mathcal{G}$ on $\mathcal{C}$. An element of the left hand side is a compatible system of maps $\mathcal{F}(U) \to \mathcal{G}(p(U))$ for all $U$ in $\mathcal{C}$. Since $\pi ^{-1}\mathcal{G}(U) = \mathcal{G}(p(U))$ by our choice of topology on $\mathcal{C}$ we see the same thing is true for the right hand side and we win.
$\square$
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