The Stacks project

Lemma 91.7.3. With notation as in (91.7.0.2) we have $Lg_{i, !} \circ g_ i^{-1} = \text{id}$ for $i = 1, 2, 3$ and hence also $L\pi _! \circ g_ i^{-1} = L\pi _{i, !}$ for $i = 1, 2, 3$.

Proof. Proof for $i = 1$. We claim the functor $\mathcal{C}_{C/B/A}$ is a fibred category over $\mathcal{C}_{B/A}$ Namely, suppose given $(P \to B, Q \to C)$ and a morphism $(P' \to B) \to (P \to B)$ of $\mathcal{C}_{B/A}$. Recall that this means we have an $A$-algebra homomorphism $P \to P'$ compatible with maps to $B$. Then we set $Q' = Q \otimes _ P P'$ with induced map to $C$ and the morphism

\[ (P' \to B, Q' \to C) \longrightarrow (P \to B, Q \to C) \]

in $\mathcal{C}_{C/B/A}$ (note reversal arrows again) is strongly cartesian in $\mathcal{C}_{C/B/A}$ over $\mathcal{C}_{B/A}$. Moreover, observe that the fibre category of $u_1$ over $P \to B$ is the category $\mathcal{C}_{C/P}$. Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}_{B/A}$. Since we have a fibred category we may apply Cohomology on Sites, Lemma 21.40.2. Thus $L_ ng_{1, !}g_1^{-1}\mathcal{F}$ is the (pre)sheaf which assigns to $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_{B/A})$ the $n$th homology of $g_1^{-1}\mathcal{F}$ restricted to the fibre category over $U$. Since these restrictions are constant the desired result follows from Lemma 91.4.4 via our identifications of fibre categories above.

The case $i = 2$. We claim $\mathcal{C}_{C/B/A}$ is a fibred category over $\mathcal{C}_{C/A}$ is a fibred category. Namely, suppose given $(P \to B, Q \to C)$ and a morphism $(Q' \to C) \to (Q \to C)$ of $\mathcal{C}_{C/A}$. Recall that this means we have a $B$-algebra homomorphism $Q \to Q'$ compatible with maps to $C$. Then

\[ (P \to B, Q' \to C) \longrightarrow (P \to B, Q \to C) \]

is strongly cartesian in $\mathcal{C}_{C/B/A}$ over $\mathcal{C}_{C/A}$. Note that the fibre category of $u_2$ over $Q \to C$ has an final (beware reversal arrows) object, namely, $(A \to B, Q \to C)$. Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}_{C/A}$. Since we have a fibred category we may apply Cohomology on Sites, Lemma 21.40.2. Thus $L_ ng_{2, !}g_2^{-1}\mathcal{F}$ is the (pre)sheaf which assigns to $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_{C/A})$ the $n$th homology of $g_1^{-1}\mathcal{F}$ restricted to the fibre category over $U$. Since these restrictions are constant the desired result follows from Cohomology on Sites, Lemma 21.39.5 because the fibre categories all have final objects.

The case $i = 3$. In this case we will apply Cohomology on Sites, Lemma 21.40.3 to $u = u_3 : \mathcal{C}_{C/B/A} \to \mathcal{C}_{C/B}$ and $\mathcal{F}' = g_3^{-1}\mathcal{F}$ for some abelian sheaf $\mathcal{F}$ on $\mathcal{C}_{C/B}$. Suppose $U = (\overline{Q} \to C)$ is an object of $\mathcal{C}_{C/B}$. Then $\mathcal{I}_ U = \mathcal{C}_{\overline{Q}/B/A}$ (again beware of reversal of arrows). The sheaf $\mathcal{F}'_ U$ is given by the rule $(P \to B, Q \to \overline{Q}) \mapsto \mathcal{F}(Q \otimes _ P B \to C)$. In other words, this sheaf is the pullback of a sheaf on $\mathcal{C}_{\overline{Q}/C}$ via the morphism $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{\overline{Q}/B/A}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{\overline{Q}/B})$. Thus Lemma 91.7.2 shows that $H_ n(\mathcal{I}_ U, \mathcal{F}'_ U) = 0$ for $n > 0$ and equal to $\mathcal{F}(\overline{Q} \to C)$ for $n = 0$. The aforementioned Cohomology on Sites, Lemma 21.40.3 implies that $Lg_{3, !}(g_3^{-1}\mathcal{F}) = \mathcal{F}$ and the proof is done. $\square$


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