ScalingStacks

Now suppose $k⩾1$ and suppose we have constructed a chain of objects and morphisms

$$M=M_0\stackrel{{\mu }_0}{⟶}{M}_1\stackrel{{\mu }_1}{⟶}{M}_2\stackrel{{\mu }_2}{⟶}{M}_3\stackrel{{\mu }_3}{⟶}\mathrm{\cdots }\stackrel{{\mu }_{k-1}}{⟶}{M}_k$$

with $M_i\in S^{{⟂}_i}$ for $1⩽i⩽k$, and where ${\mu }_i:M_i\to M_{i+1}$ is either the identity map or sits in a distinguished triangle

$${\mathrm{\Sigma }}^{-(i+1)}{S}^{(m_{i+1})}\to M_i\stackrel{{\mu }_i}{⟶}{M}_{i+1}\to {\mathrm{\Sigma }}^{-i}{S}^{(m_{i+1})}.$$

If $\text{Hom}(S,{\mathrm{\Sigma }}^{k+1}{M}_{k+1})=0$ then set $M_{k+1}=M_k$ and take ${\mu }_k:M_k\to M_{k+1}$ to be the identity map $1_{M_k}$. If not, we can choose a, possibly infinite, coproduct $S^{(m_{k+1})}$ of copies of $S$ and a nonzero morphism $S^{(m_{k+1})}\to {\mathrm{\Sigma }}^{k+1}{M}_k$ which becomes an isomorphism under $\text{Hom}(S,-)$, and then extend it to a distinguished triangle:

(3.2)

$$S^{(m_{k+1})}\to {\mathrm{\Sigma }}^{k+1}{M}_k\to {\mathrm{\Sigma }}^{k+1}{M}_{k+1}\to \mathrm{\Sigma }{S}^{(m_{k+1})}.$$

As above, an argument from the long exact sequence of Hom-sets arising from (3.2) shows that

$$\text{Hom}(S,{\mathrm{\Sigma }}^i{M}_{k+1})=0\text{ for }i=1,\mathrm{\dots },k+1.$$

The case $i=k$ follows by the injectivity of $\text{Hom}(S,S^{(m_{k+1})})\stackrel{\sim }{⟶}\text{Hom}(S,{\mathrm{\Sigma }}^{k+1}{M}_k)$; and the case $i=k+1$ by its surjectivity.