The Stacks project

Proof. 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

By Example 21.39.3 we obtain the desired restriction map

Let us denote $L_{n, p}(\mathcal{F})$ this presheaf, so that $L_ n(\mathcal{F}) = L_{n, p}(\mathcal{F})^\# $. The canonical map $\gamma : \mathcal{F} \to \mathcal{F}^+$ (Sites, Theorem 7.10.10) defines a canonical map $L_{n, p}(\mathcal{F}) \to L_{n, p}(\mathcal{F}^+)$. We have to prove this map becomes an isomorphism after sheafification.

Let us use the computation of homology given in Example 21.39.2. Denote $K_\bullet (\mathcal{F}|_{\mathcal{C}_ V})$ the complex associated to the restriction of $\mathcal{F}$ to the fibre category $\mathcal{C}_ V$. By the remarks above we obtain a presheaf $K_\bullet (\mathcal{F})$ of complexes

whose cohomology presheaves are the presheaves $L_{n, p}(\mathcal{F})$. Thus it suffices to show that

becomes an isomorphism on sheafification.

Injectivity. Let $V$ be an object of $\mathcal{D}$ and let $\xi \in K_ n(\mathcal{F})(V)$ be an element which maps to zero in $K_ n(\mathcal{F}^+)(V)$. We have to show there exists a covering $\{ V_ j \to V\} $ such that $\xi |_{V_ j}$ is zero in $K_ n(\mathcal{F})(V_ j)$. We write

with $\sigma _ i \in \mathcal{F}(U_{i, 0})$. We arrange it so that each sequence of morphisms $U_ n \to \ldots \to U_0$ of $\mathcal{C}_ V$ occurs are most once. Since the sums in the definition of the complex $K_\bullet $ are direct sums, the only way this can map to zero in $K_\bullet (\mathcal{F}^+)(V)$ is if all $\sigma _ i$ map to zero in $\mathcal{F}^+(U_{i, 0})$. By construction of $\mathcal{F}^+$ there exist coverings $\{ U_{i, 0, j} \to U_{i, 0}\} $ such that $\sigma _ i|_{U_{i, 0, j}}$ is zero. By our construction of the topology on $\mathcal{C}$ we can write $U_{i, 0, j} \to U_{i, 0}$ as the pullback (Categories, Definition 4.33.6) of some morphisms $V_{i, j} \to V$ and moreover each $\{ V_{i, j} \to V\} $ is a covering. Choose a covering $\{ V_ j \to V\} $ dominating each of the coverings $\{ V_{i, j} \to V\} $. Then it is clear that $\xi |_{V_ j} = 0$.

Surjectivity. Proof omitted. Hint: Argue as in the proof of injectivity. $\square$

Proof. Consider the sequence of functors $\mathcal{F} \mapsto L_ n(\mathcal{F})$ from $\textit{PAb}(\mathcal{C}) \to \textit{Ab}(\mathcal{C})$. Since for each $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ the sequence of functors $H_ n(\mathcal{C}_ V, - )$ forms a $\delta $-functor so do the functors $\mathcal{F} \mapsto L_ n(\mathcal{F})$. Our goal is to show these form a universal $\delta $-functor. In order to do this we construct some abelian presheaves on which these functors vanish.

For $U' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ consider the abelian presheaf $\mathcal{F}_{U'} = j_{U'!}^{\textit{PAb}}\mathbf{Z}_{U'}$ (Modules on Sites, Remark 18.19.7). Recall that

If $U$ lies over $V = p(U)$ in $\mathcal{D})$ and $U'$ lies over $V' = p(U')$ then any morphism $a : U \to U'$ factors uniquely as $U \to h^*U' \to U'$ where $h = p(a) : V \to V'$ (see Categories, Definition 4.33.6). Hence we see that

where $j_{h^*U'} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ V/h^*U') \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ V)$ is the localization morphism. The sheaves $j_{h^*U'!}\mathbf{Z}_{h^*U'}$ have vanishing higher homology groups (see Example 21.39.2). We conclude that $L_ n(\mathcal{F}_{U'}) = 0$ for all $n > 0$ and all $U'$. It follows that any abelian presheaf $\mathcal{F}$ is a quotient of an abelian presheaf $\mathcal{G}$ with $L_ n(\mathcal{G}) = 0$ for all $n > 0$ (Modules on Sites, Lemma 18.28.8). Since $L_ n(\mathcal{F}) = L_ n(\mathcal{F}^\# )$ we see that the same thing is true for abelian sheaves. Thus the sequence of functors $L_ n(-)$ is a universal delta functor on $\textit{Ab}(\mathcal{C})$ (Homology, Lemma 12.12.4). Since we have agreement with $H^{-n}(L\pi _!(-))$ for $n = 0$ by Lemma 21.38.8 we conclude by uniqueness of universal $\delta $-functors (Homology, Lemma 12.12.5) and Derived Categories, Lemma 13.16.6. $\square$

Proof. Say $p(U) = V$. The category $\mathcal{I}_ U$ is the category of pairs $(U', \varphi )$ where $\varphi : U \to u(U')$ is a morphism of $\mathcal{C}$ with $p(\varphi ) = \text{id}_ V$, i.e., $\varphi $ is a morphism of the fibre category $\mathcal{C}_ V$. Morphisms $(U'_1, \varphi _1) \to (U'_2, \varphi _2)$ are given by morphisms $a : U'_1 \to U'_2$ of the fibre category $\mathcal{C}'_ V$ such that $\varphi _2 = u(a) \circ \varphi _1$. The presheaf $\mathcal{F}'_ U$ sends $(U', \varphi )$ to $\mathcal{F}'(U')$. We will construct the restriction mappings below.

Choose a factorization

of $u$ as in Categories, Lemma 4.33.14. Then $g_! = g''_! \circ g'_!$ and similarly for derived functors. On the other hand, the functor $g'_!$ is exact, see Modules on Sites, Lemma 18.16.6. Thus we get $Lg_!(\mathcal{F}') = Lg''_!(\mathcal{F}'')$ where $\mathcal{F}'' = g'_!\mathcal{F}'$. Note that $\mathcal{F}'' = h^{-1}\mathcal{F}'$ where $h : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}'') \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}')$ is the morphism of topoi associated to $w$, see Sites, Lemma 7.23.1. The functor $u''$ turns $\mathcal{C}''$ into a fibred category over $\mathcal{C}$, hence Lemma 21.40.2 applies to the computation of $L_ ng''_!$. The result follows as the construction of $\mathcal{C}''$ in the proof of Categories, Lemma 4.33.14 shows that the fibre category $\mathcal{C}''_ U$ is equal to $\mathcal{I}_ U$. Moreover, $h^{-1}\mathcal{F}'|_{\mathcal{C}''_ U}$ is given by the rule described above (as $w$ is continuous and cocontinuous by Stacks, Lemma 8.10.3 so we may apply Sites, Lemma 7.21.5). $\square$