The Stacks project

Proof. If $\mathcal{C}$ has an initial object, then $\pi _!$ is computed by evaluating on this object and the statement is clear. If $\mathcal{C}$ has a final object, then $R\pi _*$ is computed by evaluating on this object, hence $R\pi _* \circ \pi ^{-1} \cong \text{id}$ on $D(\textit{Ab})$, resp. $D(B)$. This implies that $\pi ^{-1} : D(\textit{Ab}) \to D(\mathcal{C})$, resp. $\pi ^{-1} : D(B) \to D(\underline{B})$ is fully faithful, see Categories, Lemma 4.24.4. Then the same lemma implies that $L\pi _! \circ \pi ^{-1} = \text{id}$ as desired. $\square$

Proof. Both functors are right adjoint to the obvious functor $D(B') \to D(\underline{B})$. $\square$

Proof. As $L\pi _!$ agrees for modules and abelian sheaves by Lemma 21.38.5 it suffices to prove this when $\mathcal{F}$ is an abelian sheaf. For $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ the abelian sheaf $j_{U!}\mathbf{Z}_ U$ is a projective object of $\textit{Ab}(\mathcal{C})$ since $\mathop{\mathrm{Hom}}\nolimits (j_{U!}\mathbf{Z}_ U, \mathcal{F}) = \mathcal{F}(U)$ and taking sections is an exact functor as the topology is chaotic. Every abelian sheaf is a quotient of a direct sum of $j_{U!}\mathbf{Z}_ U$ by Modules on Sites, Lemma 18.28.8. Thus we can compute $L\pi _!(\mathcal{F})$ by choosing a resolution

whose terms are direct sums of sheaves of the form above and taking $L\pi _!(\mathcal{F}) = \pi _!(\mathcal{G}^\bullet )$. Consider the double complex $A^{\bullet , \bullet } = \mathcal{G}^\bullet (U_\bullet )$. The map $\mathcal{G}^0 \to \mathcal{F}$ gives a map of complexes $A^{0, \bullet } \to \mathcal{F}(U_\bullet )$. Since $\pi _!$ is computed by taking the colimit over $\mathcal{C}^{opp}$ (Lemma 21.38.8) we see that the two compositions $\mathcal{G}^ m(U_1) \to \mathcal{G}^ m(U_0) \to \pi _!\mathcal{G}^ m$ are equal. Thus we obtain a canonical map of complexes

To prove the lemma it suffices to show that the complexes

are exact, see Homology, Lemma 12.25.4. Since the sheaves $\mathcal{G}^ m$ are direct sums of the sheaves $j_{U!}\mathbf{Z}_ U$ we reduce to $\mathcal{G} = j_{U!}\mathbf{Z}_ U$. The complex $j_{U!}\mathbf{Z}_ U(U_\bullet )$ is the complex of abelian groups associated to the free $\mathbf{Z}$-module on the simplicial set $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U_\bullet , U)$ which we assumed to be homotopy equivalent to a singleton. We conclude that

is a homotopy equivalence of abelian groups hence a quasi-isomorphism (Simplicial, Remark 14.26.4 and Lemma 14.27.1). This finishes the proof since $\pi _!j_{U!}\mathbf{Z}_ U = \mathbf{Z}$ as was shown in the proof of Lemma 21.38.5. $\square$

Proof. Follows immediately from Lemma 21.39.7 and the fact that $g^{-1}$ is given by precomposing with $u$. $\square$

Proof. As both sides commute with colimits, it suffices to prove this for $K_1 = j_{U!}\underline{B}_ U$ and $K_2 = j_{V!}\underline{B}_ V$ for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_1)$ and $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_2)$. See construction of $L\pi _!$ in Lemma 21.37.2. In this case

Verification omitted. Hence the result follows as both the left and the right hand side of the formula of the lemma evaluate to $B$, see construction of $L\pi _!$ in Lemma 21.37.2. $\square$

Proof. Consider the diagram of categories and functors

where $u$ is the diagonal functor and $u_ i$ are the projection functors. This gives morphisms of ringed topoi $g$, $g_1$, $g_2$. For any object $(U_1, U_2)$ of $\mathcal{C}$ we have

which is homotopy equivalent to a point by Simplicial, Lemma 14.26.10. Thus Lemma 21.39.8 gives $L\pi _!(g^{-1}K) = L(\pi \times \pi )_!(K)$ for any $K$ in $D(\mathcal{C} \times \mathcal{C}, B)$. Take $K = g_1^{-1}K_1 \otimes _ B^\mathbf {L} g_2^{-1}K_2$. Then $g^{-1}K = K_1 \otimes ^\mathbf {L}_{\underline{B}} K_2$ because $g^{-1} = g^* = Lg^*$ commutes with derived tensor product (Lemma 21.18.4). To finish we apply Lemma 21.39.9. $\square$

Proof. Note: in this proof $L\pi _!$ denotes the left derived functor of $\pi _!$ on abelian sheaves. Since $L\pi _!$ commutes with colimits, it suffices to prove this for bounded above complexes of $\mathcal{O}$-modules (compare with argument of Derived Categories, Proposition 13.29.2 or just stick to bounded above complexes). Every such complex is quasi-isomorphic to a bounded above complex whose terms are direct sums of $j_{U!}\mathcal{O}_ U$ with $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, see Modules on Sites, Lemma 18.28.8. Thus it suffices to prove the lemma for $j_{U!}\mathcal{O}_ U$. By assumption

is a simplicial set homotopy equivalent to the constant simplicial set on a singleton. Set $P_ n = \mathcal{O}(U_ n)$ and $P'_ n = \mathcal{O}'(U_ n)$. Observe that the complex associated to the simplicial abelian group

computes $L\pi _!(j_{U!}\mathcal{O}_ U)$ by Lemma 21.39.7. Since $j_{U!}\mathcal{O}_ U$ is a flat $\mathcal{O}$-module we have $j_{U!}\mathcal{O}_ U \otimes ^\mathbf {L}_\mathcal {O} \mathcal{O}' = j_{U!}\mathcal{O}'_ U$ and $L\pi _!$ of this is computed by the complex associated to the simplicial abelian group

As the rule which to a simplicial set $T_\bullet$ associated the simplicial abelian group with terms $\bigoplus _{t \in T_ n} P_ n$ is a functor, we see that $X_\bullet \to P_\bullet$ is a homotopy equivalence of simplicial abelian groups. Similarly, the rule which to a simplicial set $T_\bullet$ associates the simplicial abelian group with terms $\bigoplus _{t \in T_ n} P'_ n$ is a functor. Hence $X'_\bullet \to P'_\bullet$ is a homotopy equivalence of simplicial abelian groups. By assumption $P_\bullet \to P'_\bullet$ is a quasi-isomorphism (since $P_\bullet$, resp. $P'_\bullet$ computes $L\pi _!\mathcal{O}$, resp. $L\pi _!\mathcal{O}'$ by Lemma 21.39.7). We conclude that $X_\bullet$ and $X'_\bullet$ are quasi-isomorphic as desired. $\square$