Proposition 5.3.6 (0P70) · ScalingStacks

0P70

Proposition 5.3.6. The pair $(E,\tau)$ defines a $2$ -representation on $A$ and $\Phi$ induces a isomorphism of $2$ -representations $\Delta_{\sigma}(B\operatorname{\!-diff}\nolimits)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}(\Delta_{\sigma}B)\operatorname{\!-diff}\nolimits$ . If $E_{2}$ is right finite, then $E$ is right finite.

0P71

Proof. The fact that $\tau$ defines an endomorphism of $(A,A)$ -bimodules of $E^{2}$ satisfying the appropriate relations follows from the fact that it agrees with the endomorphism of ${\mathcal{E}}^{2}$ defining the $2$ -representation on $\Delta_{\sigma}(B\operatorname{\!-diff}\nolimits)$ . We deduce that $(E,\tau)$ is a $2$ -representation on $A$ and $\Phi$ is a morphism of $2$ -representations.

Note that $E$ is finitely generated and projective as a (non-differential) $A^{\operatorname{opp}\nolimits}$ -module if $E_{1}$ and $E_{2}$ are finitely generated and projective $B^{\operatorname{opp}\nolimits}$ -modules. ∎