The Stacks project

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. $\square$