ScalingStacks

Proof. We briefly sketch the argument of [L1, Proposition 5.5.5]. The derived global sections functor $\Gamma$ preserves finite colimits. Thus it suffices to show $\Gamma$ preserves small coproducts: we must check that the natural map $\coprod_{\alpha}\Gamma(M_{\alpha})\rightarrow\Gamma(\coprod_{\alpha}M_{\alpha})$ is an equivalence, i.e., that the induced map on homotopy groups $\coprod_{\alpha}\pi_{*}\Gamma(M_{\alpha})\rightarrow\pi_{*}\Gamma(\coprod_{\alpha}M_{\alpha})$ is an equivalence. Since $\pi_{*}$ is colimit preserving, it suffices to check that the individual terms $\pi_{i}\Gamma$ each preserve small coproducts.

This is shown in two steps. First, one checks that for $M\in\qc(X)$ concentrated in a single degree, there exists $m$ such that $\pi_{n}\Gamma M$ is zero for all $n<m$. Thus to establish the assertion, it suffices to work in the subcategory $M\in(\qc(X))^{\leq n+m}_{\geq n}$ for which $\pi_{*}\Gamma M$ is concentrated in bounded degrees.

Next, one chooses a finite affine cover $U\rightarrow X$ giving the usual simplicial object $U_{*}\rightarrow X$, and thus an identification $\Gamma M\simeq\lim(M|_{U_{q}})$. The resulting Bousfield-Kan, or C̆ech, spectral sequence has $E_{1}$-term given by the $\pi_{p}(M|_{U_{q}})$, and converges to $\pi_{*}\Gamma M$. (The construction of the $E_{1}$-term evidently commutes with coproducts in $\qc(X)$, since pullback to an affine is colimit preserving.) Using the previous step, only finitely many terms in the spectral sequence may involve differentials affecting a particular group $\pi_{i}\Gamma M$. Therefore we have expressed $\pi_{i}\Gamma M$ as a finite limit of terms which preserved colimits in $M$ and so the result follows. ∎