Lemma 85.33.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. If $a : U \to X$ is an fppf hypercovering of $X$, then for $K \in D^+(X_{\acute{e}tale})$

\[ K \to Ra_*(a^{-1}K) \]

is an isomorphism. Here $a : \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ is as in Section 85.32.

**Proof.**
Consider the diagram of Lemma 85.33.1. Observe that $Rh_{n, *}h_ n^{-1}$ is the identity functor on $D^+(U_{n, {\acute{e}tale}})$ by More on Cohomology of Spaces, Lemma 84.6.2. Hence $Rh_*h^{-1}$ is the identity functor on $D^+(U_{\acute{e}tale})$ by Lemma 85.5.3. We have

\begin{align*} Ra_*(a^{-1}K) & = Ra_*Rh_*h^{-1}a^{-1}K \\ & = Rh_{-1, *}Ra_{fppf, *}a_{fppf}^{-1}(h_{-1})^{-1}K \\ & = Rh_{-1, *}(h_{-1})^{-1}K \\ & = K \end{align*}

The first equality by the discussion above, the second equality because of the commutativity of the diagram in Lemma 85.25.1, the third equality by Lemma 85.21.2 as $U$ is a hypercovering of $X$ in $(\textit{Spaces}/S)_{fppf}$, and the last equality by the already used More on Cohomology of Spaces, Lemma 84.6.2.
$\square$

## Comments (0)