Section 18.41 (0796): Lower shriek for modules

18.41 Lower shriek for modules

In this section we extend the construction of $g_!$ discussed in Section 18.16 to the case of sheaves of modules.

Lemma 18.41.1. Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous and cocontinuous functor between sites. Denote $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ the associated morphism of topoi. Let $\mathcal{O}_\mathcal {D}$ be a sheaf of rings on $\mathcal{D}$ . Set $\mathcal{O}_\mathcal {C} = g^{-1}\mathcal{O}_\mathcal {D}$ . Hence $g$ becomes a morphism of ringed topoi with $g^* = g^{-1}$ . In this case there exists a functor

$g_! : \textit{Mod}(\mathcal{O}_\mathcal {C}) \longrightarrow \textit{Mod}(\mathcal{O}_\mathcal {D})$

which is left adjoint to $g^*$ .

Proof. Let $U$ be an object of $\mathcal{C}$ . For any $\mathcal{O}_\mathcal {D}$ -module $\mathcal{G}$ we have

$\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {C}}(j_{U!}\mathcal{O}_ U, g^{-1}\mathcal{G}) & = g^{-1}\mathcal{G}(U) \\ & = \mathcal{G}(u(U)) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {D}}(j_{u(U)!}\mathcal{O}_{u(U)}, \mathcal{G}) \end{align*}$

because $g^{-1}$ is described by restriction, see Sites, Lemma 7.21.5. Of course a similar formula holds a direct sum of modules of the form $j_{U!}\mathcal{O}_ U$ . By Homology, Lemma 12.29.6 and Lemma 18.28.8 we see that $g_!$ exists. $\square$

Remark 18.41.2. Warning! Let $u : \mathcal{C} \to \mathcal{D}$ , $g$ , $\mathcal{O}_\mathcal {D}$ , and $\mathcal{O}_\mathcal {C}$ be as in Lemma 18.41.1. In general it is not the case that the diagram

$\xymatrix{ \textit{Mod}(\mathcal{O}_\mathcal {C}) \ar[r]_{g_!} \ar[d]_{forget} & \textit{Mod}(\mathcal{O}_\mathcal {D}) \ar[d]^{forget} \\ \textit{Ab}(\mathcal{C}) \ar[r]^{g^{Ab}_!} & \textit{Ab}(\mathcal{D}) }$

commutes (here $g^{Ab}_!$ is the one from Lemma 18.16.2). There is a transformation of functors

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

From the proof of Lemma 18.41.1 we see that this is an isomorphism if and only if $g^{Ab}_!j_{U!}\mathcal{O}_ U \to g_!j_{U!}\mathcal{O}_ U$ is an isomorphism for all objects $U$ of $\mathcal{C}$ . Since we have $g_!j_{U!}\mathcal{O}_ U = j_{u(U)!}\mathcal{O}_{u(U)}$ this holds if and only if

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

is an isomorphism for all objects $U$ of $\mathcal{C}$ . Note that for such a $U$ we obtain a commutative diagram

$\xymatrix{ \mathcal{C}/U \ar[r]_-{u'} \ar[d]_{j_ U} & \mathcal{D}/u(U) \ar[d]^{j_{u(U)}} \\ \mathcal{C} \ar[r]^ u & \mathcal{D} }$

of cocontinuous functors of sites, see Sites, Lemma 7.28.4 and therefore $g^{Ab}_!j_{U!} = j_{u(U)!}(g')^{Ab}_!$ where $g' : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/u(U))$ is the morphism of topoi induced by the cocontinuous functor $u'$ . Hence we see that $g_! = g^{Ab}_!$ if the canonical map

18.41.2.1

$\begin{equation} \label{sites-modules-equation-compare-on-localizations} (g')^{Ab}_!\mathcal{O}_ U \longrightarrow \mathcal{O}_{u(U)} \end{equation}$

is an isomorphism for all objects $U$ of $\mathcal{C}$ .

The following two results are of a slightly different nature.

Lemma 18.41.3. 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

$f$ , $f'$ , $g$ , and $g'$ correspond to cocontinuous functors $u$ , $u'$ , $v$ , and $v'$ as in Sites, Lemma 7.21.1,
$v \circ u' = u \circ v'$ ,
$v$ and $v'$ are continuous as well as cocontinuous,
for any object $V'$ of $\mathcal{D}'$ the functor ${}^{u'}_{V'}\mathcal{I} \to {}^{\ \ \ u}_{v(V')}\mathcal{I}$ given by $v$ is cofinal, and
$g^{-1}\mathcal{O}_{\mathcal{D}} = \mathcal{O}_{\mathcal{D}'}$ and $(g')^{-1}\mathcal{O}_{\mathcal{C}} = \mathcal{O}_{\mathcal{C}'}$ .

Then we have $f'_* \circ (g')^* = g^* \circ f_*$ and $g'_! \circ (f')^{-1} = f^{-1} \circ g_!$ on modules.

Proof. We have $(g')^*\mathcal{F} = (g')^{-1}\mathcal{F}$ and $g^*\mathcal{G} = g^{-1}\mathcal{G}$ because of condition (5). Thus the first equality follows immediately from the corresponding equality in Sites, Lemma 7.28.6. Since the left adjoint functors $g_!$ and $g'_!$ to $g^*$ and $(g')^*$ exist by Lemma 18.41.1 we see that the second equality follows by uniqueness of adjoint functors. $\square$

Lemma 18.41.4. Consider a commutative diagram

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

$u$ and $u'$ are continuous and give rise to the morphisms $f$ and $f'$ ,
$v$ and $v'$ are cocontinuous giving rise to the morphisms $g$ and $g'$ ,
$u \circ v = v' \circ u'$ ,
$v$ and $v'$ are continuous as well as cocontinuous, and
$g^{-1}\mathcal{O}_{\mathcal{D}} = \mathcal{O}_{\mathcal{D}'}$ and $(g')^{-1}\mathcal{O}_{\mathcal{C}} = \mathcal{O}_{\mathcal{C}'}$ .

Then $f'_* \circ (g')^* = g^* \circ f_*$ and $g'_! \circ (f')^{-1} = f^{-1} \circ g_!$ on modules.

Proof. We have $(g')^*\mathcal{F} = (g')^{-1}\mathcal{F}$ and $g^*\mathcal{G} = g^{-1}\mathcal{G}$ because of condition (5). Thus the first equality follows immediately from the corresponding equality in Sites, Lemma 7.28.7. Since the left adjoint functors $g_!$ and $g'_!$ to $g^*$ and $(g')^*$ exist by Lemma 18.41.1 we see that the second equality follows by uniqueness of adjoint functors. $\square$

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18.41 Lower shriek for modules

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