ScalingStacks

Proof. We have $acT_{i}b=aT_{i+1}cb$ for $i\in\{1,\ldots,s-2\}$. This shows the first statement of the lemma.

We have now a morphism of differential graded algebras and of $(H_{s},H_{s})$-bimodules $f^{\prime}:T_{H_{s}}(M_{s})\to\hat{H}_{s}^{+}$ induced by the morphism $M_{s}\to\hat{H}_{s}^{+}$. We have

$$f^{\prime}((1\otimes 1)\otimes(1\otimes T_{s-1}))=c^{2}T_{s-1}=T_{1}c^{2}=f^{\prime}((T_{1}\otimes 1)\otimes(1\otimes 1)),$$

hence $f^{\prime}(\kappa)=0$. So, $f$ induces a morphism of algebras $f:T_{H_{s}}(M_{s})/(\kappa)\to\hat{H}_{s}^{+}$.

On the other hand, $\hat{H}_{s}^{+}$ is the free algebra generated by $H_{s}$ and $c$ with the relations $cT_{i}=T_{i+1}c$ for $i\in\{1,\ldots,s-2\}$ and $c^{2}T_{s-1}=T_{1}c^{2}$ (Proposition 3.2.9). Since $T_{i+1}\otimes 1=1\otimes T_{i}$ in $M_{s}$ for $i\in\{1,\ldots,s-2\}$ and $(1\otimes 1)\otimes(1\otimes T_{s-1})=(T_{1}\otimes 1)\otimes(1\otimes 1)$ in $M_{s}\otimes M_{s}$, we deduce that there is a morphism of algebras $g:\hat{H}_{s}^{+}\to T_{H_{s}}(M_{s})/(\kappa),\ T_{i}\mapsto T_{i},\ c\mapsto 1\otimes 1$. The morphisms $f$ and $g$ are inverse and we are done. ∎