ScalingStacks

Proof: Applying the functor $\text{Hom}(S,-)$ to distinguished triangle (3.4),

$${\mathrm{\Sigma }}^{-(k+1)}{S}^{(m_{k+1})}\to M_k\stackrel{{\mu }_k}{⟶}{M}_{k+1}\to {\mathrm{\Sigma }}^{-k}{S}^{(m_{k+1})},$$

for in the proof of Proposition 3.2 shows that

$$\text{Hom}(S,{\mathrm{\Sigma }}^i\mu ):\text{Hom}(S,{\mathrm{\Sigma }}^i{M}_k)\to \text{Hom}(S,{\mathrm{\Sigma }}^i{M}_{k+1})$$

is an isomorphism for all . The isomorphism for $i=k$ follows by the fact that the morphism $S^{(m_{k+1})}\to {\mathrm{\Sigma }}^{k+1}{M}_k$ in (3.4) is constructed to be an isomorphism under $\text{Hom}(S,-)$. Hence the composite $\mu ={\mu }_{n-1}\circ \mathrm{\cdots }\circ {\mu }_1\circ {\mu }_0$ is an isomorphism under $\text{Hom}(S,{\mathrm{\Sigma }}^i-)$ for all . $\mathrm{\square }$