Lemma 3.3 (0MUI) · ScalingStacks

0MUI

Lemma 3.3. Suppose further that $S$ is a simply connected corigid object of $\mathcal{T}$ . Then, for the $S^{\perp_{n}}$ -preenvelope, $\mu:M\rightarrow\bar{M}$ , obtained in Proposition 3.2, we have that

\textnormal{Hom}(S,\Sigma^{i}\mu):\textnormal{Hom}(S,\Sigma^{i}M)\rightarrow% \textnormal{Hom}(S,\Sigma^{i}\bar{M})

is an isomorphism for all $i<1$ .

0MUJ

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

\Sigma^{-(k+1)}S^{(m_{k+1})}\rightarrow M_{k}\stackrel{{\scriptstyle\mu_{k}}}{% {\longrightarrow}}M_{k+1}\rightarrow\Sigma^{-k}S^{(m_{k+1})},

for $0\leqslant k<n$ in the proof of Proposition 3.2 shows that

\textnormal{Hom}(S,\Sigma^{i}\mu):\textnormal{Hom}(S,\Sigma^{i}M_{k})% \rightarrow\textnormal{Hom}(S,\Sigma^{i}M_{k+1})

is an isomorphism for all $i<k+1$ . The isomorphism for $i=k$ follows by the fact that the morphism $S^{(m_{k+1})}\rightarrow\Sigma^{k+1}M_{k}$ in (3.4) is constructed to be an isomorphism under $\textnormal{Hom}(S,-)$ . Hence the composite $\mu=\mu_{n-1}\circ\cdots\circ\mu_{1}\circ\mu_{0}$ is an isomorphism under $\textnormal{Hom}(S,\Sigma^{i}-)$ for all $i<1$ . $\Box$