Proposition 9.45 (0NQH) · ScalingStacks

0NQH

Proposition 9.45. An $F^{\infty}_{R}$ -module $M$ is in the full subcategory ${\mathcal{C}}\subseteq\Psi(F^{\infty}_{R})$ spanned by the local objects if and only if $i^{*}M\simeq 0$ in $\Psi(F_{R})$ . Similarly, a map of $F^{\infty}_{R}$ -modules $f\colon M\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}M^{\prime}$ is a local equivalence if and only if the cofiber of $f$ lies in the essential image of $i_{!}$ .

0NQI

Proof. The first claim follows from the fact that $i^{*}M\simeq 0$ if and only if for all $F_{R}$ -modules $N$ , $\mathrm{Map}_{F^{\infty}_{R}}(i_{!}N,M)\simeq 0$ . In turn, this holds if and only if for any map of $F^{\infty}_{R}$ -modules $Q\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}P$ with cofiber of the form $i_{!}N$ ,

\mathrm{Map}_{F^{\infty}_{R}}(P,M)\simeq\mathrm{Map}_{F^{\infty}_{R}}(Q,M).

The second claim follows from the fact that, if the cofiber of $f$ lies in the essential image of $i_{!}$ , then for any local object $L$ , $\mathrm{Map}_{F^{\infty}_{R}}(M^{\prime},L)\simeq\mathrm{Map}_{F^{\infty}_{R}}(M,L)$ . ∎