Corollary 9.46 (0NQJ) · ScalingStacks

0NQJ

Corollary 9.46. Let $G$ denote the cofiber of the counit $i_{!}i^{*}R^{\lor\infty}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}R^{\lor\infty}$ in $\Psi(F^{\infty}_{R})$ . Then $G$ lies in the full subcategory ${\mathcal{C}}\subseteq\Psi(F^{\infty}_{R})$ , i.e. $G$ is a local object, and the map $R^{\lor\infty}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}G$ is a local equivalence. Furthermore, $G$ is a compact generator of ${\mathcal{C}}$ .

0NQK

Proof. By the previous proposition, $G$ is a local object, and the cofiber

\Sigma i_{!}i^{*}R^{\lor\infty}\simeq i_{!}\Sigma i^{*}R^{\lor\infty}

of $R^{\lor\infty}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}G$ is in the image of $i_{!}$ . $G$ is compact because $R^{\lor\infty}$ is a compact generator of $F^{\infty}_{R}$ and the functor $\Psi(F^{\infty}_{R})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ preserves compact objects [70, 2.9]. ∎