The Stacks project

Lemma 21.35.7. 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 exact1,

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 21.35.1 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 18.40.4 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 13.30.9 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 18.40.4 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 fibre products and equalizers exist in $\mathcal{C}'$ and $v'$ commutes with them, see Modules on Sites, Lemma 18.16.3.

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