ScalingStacks

Let $(M,\varsigma)$ be an object of $\Delta_{E\otimes_{B}-}(B\operatorname{\!-diff}\nolimits)$. The action of $T_{B}(E_{0,1}E_{1,0})$ on $M$ vanishes on $K_{i}$ for all $i$, hence defines an action of $A$ on $M$. This gives a fully faithful differential functor $\Delta_{E\otimes_{B}-}(B\operatorname{\!-diff}\nolimits)\to(\Delta_{E}(B))\operatorname{\!-diff}\nolimits$. If the canonical injective morphism of differential $(B,B)$-bimodules

(5.2.1)

$$(E_{0,1}E_{1,0})^{i}/K_{i}\to E_{i,i}/((T_{r}\otimes 1)x-(1\otimes T_{r})x)_{x\in E_{i,i},\ 1\leq r<i}$$

is a split injection for all $i\geq 1$, then the functor above is an isomorphism

$$\Delta_{E\otimes_{B}-}(B\operatorname{\!-diff}\nolimits)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}(\Delta_{E}(B))\operatorname{\!-diff}\nolimits.$$