The Stacks project

Proof. The cartesian $\mathcal{O}$-modules form a weak Serre subcategory by Lemma 85.12.6 (the required hypotheses hold by the discussion in Remark 85.16.5). Since the restriction functor $g_ n^* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_ n)$ are exact, it follows that $\mathcal{A}_{total}$ is a weak Serre subcategory.

Let us show that $a^* : \mathcal{A} \to \mathcal{A}_{total}$ is an equivalence of categories with inverse given by $La_*$. We already know that $La_*a^*\mathcal{F} = \mathcal{F}$ by the bounded version (Lemma 85.18.4). It is clear that $a^*\mathcal{F}$ is in $\mathcal{A}_{total}$ for $\mathcal{F}$ in $\mathcal{A}$. Conversely, assume that $\mathcal{G} \in \mathcal{A}_{total}$. Because $\mathcal{G}$ is cartesian we see that $\mathcal{G} = a^*\mathcal{F}$ for some $\mathcal{O}_\mathcal {C}$-module $\mathcal{F}$ by Lemma 85.18.1. We want to show that $\mathcal{F}$ is in $\mathcal{A}$. Take $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. We have to show that the restriction of $\mathcal{F}$ to $\mathcal{C}/U$ is in $\mathcal{A}_ U$. As usual, write $K_0 = \{ U_{0, i}\} _{i \in I_0}$. Since $K$ is a hypercovering, the map $\coprod _{i \in I_0} h_{U_{0, i}} \to *$ becomes surjective after sheafification. This implies there is a covering $\{ U_ j \to U\} _{j \in J}$ and a map $\tau : J \to I_0$ and for each $j \in J$ a morphism $\varphi _ j : U_ j \to U_{0, \tau (j)}$. Since $\mathcal{G}_0 = a_0^*\mathcal{F}$ we find that the restriction of $\mathcal{F}$ to $\mathcal{C}/U_ j$ is equal to the restriction of the $\tau (j)$th component of $\mathcal{G}_0$ to $\mathcal{C}/U_ j$ via the morphism $\varphi _ j : U_ j \to U_{0, \tau (i)}$. Hence by (1) we find that $\mathcal{F}|_{\mathcal{C}/U_ j}$ is in $\mathcal{A}_{U_ j}$ and in turn by (2) we find that $\mathcal{F}|_{\mathcal{C}/U}$ is in $\mathcal{A}_ U$.

In particular the statement of the lemma makes sense. The lemma now follows from Cohomology on Sites, Lemma 21.28.6. Assumption (1) is clear (see Remark 85.16.5). Assumptions (2) and (3) we proved in the preceding paragraph. Assumption (4) is immediate from (3). For assumption (5) let $\mathcal{B}_{total}$ be the set of objects $U/U_{n, i}$ of the site $(\mathcal{C}/K)_{total}$ such that $U \in \mathcal{B}$ where $\mathcal{B}$ is as in (3). Here we use the description of the site $(\mathcal{C}/K)_{total}$ given in Section 85.16. Moreover, we set $\text{Cov}_{U/U_{n, i}}$ equal to $\text{Cov}_ U$ and $d_{U/U_{n, i}}$ equal $d_ U$ where $\text{Cov}_ U$ and $d_ U$ are given to us by (3). Then we claim that condition (5) holds with these choices. This follows immediately from Lemma 85.16.3 and the fact that $\mathcal{F} \in \mathcal{A}_{total}$ implies $\mathcal{F}_ n \in \mathcal{A}_{K_ n}$ and hence $\mathcal{F}_{n, i} \in \mathcal{A}_{U_{n, i}}$. (The reader who worries about the difference between cohomology of abelian sheaves versus cohomology of sheaves of modules may consult Cohomology on Sites, Lemma 21.12.4.) $\square$