Lemma 115.8.7. In Simplicial Spaces, Situation 85.3.3 let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in Simplicial Spaces, Remark 85.4.1. Suppose given strictly full weak Serre subcategories

$\mathcal{A} \subset \textit{Ab}(\mathcal{D}),\quad \mathcal{A}_ n \subset \textit{Ab}(\mathcal{C}_ n)$

Then

1. the collection of abelian sheaves $\mathcal{F}$ on $\mathcal{C}_{total}$ whose restriction to $\mathcal{C}_ n$ is in $\mathcal{A}_ n$ for all $n$ is a strictly full weak Serre subcategory $\mathcal{A}_{total} \subset \textit{Ab}(\mathcal{C}_{total})$.

If $a_ n^{-1}$ sends $\mathcal{A}$ into $\mathcal{A}_ n$ for all $n$, then

1. $a^{-1}$ sends $\mathcal{A}$ into $\mathcal{A}_{total}$ and

2. $a^{-1}$ sends $D_\mathcal {A}(\mathcal{D})$ into $D_{\mathcal{A}_{total}}(\mathcal{C}_{total})$.

If $R^ qa_{n, *}$ sends $\mathcal{A}_ n$ into $\mathcal{A}$ for all $n, q$, then

1. $R^ qa_*$ sends $\mathcal{A}_{total}$ into $\mathcal{A}$ for all $q$, and

2. $Ra_*$ sends $D_{\mathcal{A}_{total}}^+(\mathcal{C}_{total})$ into $D_\mathcal {A}^+(\mathcal{D})$.

Proof. The only interesting assertions are (4) and (5). Part (4) follows from the spectral sequence in Simplicial Spaces, Lemma 85.9.3 and Homology, Lemma 12.24.11. Then part (5) follows by considering the spectral sequence associated to the canonical filtration on an object $K$ of $D_{\mathcal{A}_{total}}^+(\mathcal{C}_{total})$ given by truncations. We omit the details. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).