Subsection 5.4.1. (0PG8) · ScalingStacks

5.4.1. Algebra

Let $B$ be a differential algebra endowed with two $2$ -representations $(F_{1},\tau_{1})$ and $(E_{2},\tau_{2})$ together with a closed morphism $\lambda:F_{1}E_{2}\to E_{2}F_{1}$ such that the diagrams (4.2.1) commute.

We define the differential algebra

A=\Delta_{\lambda}(B)=\bigoplus_{i\geq 0}(E_{2}^{i}F_{1}^{i})/((T_{r}\otimes 1)x-(1\otimes T_{r})x)_{x\in E_{2}^{i}F_{1}^{i},\ 1\leq r<i}.

Its multiplication is given by the maps $\mu_{i,j}=\mu_{(i,i),(j,j)}E_{2}^{i}F_{1}^{i}E_{2}^{j}F_{1}^{j}\to E_{2}^{i+j}F_{1}^{i+j}$ defined in §4.2.1.

Given $M$ a differential $A$ -module and given $i\geq 1$ , we have differential $B$ -module maps $\varsigma_{i}:E_{2}^{i}F_{1}^{i}\otimes_{B}M\to M$ . These make $(M,(\varsigma_{i})_{i})$ into an object of $\Delta_{\lambda}(B\operatorname{\!-diff}\nolimits)$ and provides an isomorphism of differential categories $\Delta_{\lambda}(B)\operatorname{\!-diff}\nolimits\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\Delta_{\lambda}(B\operatorname{\!-diff}\nolimits)$ .

0P73

Remark 5.4.1. As in Remark 4.4.3, we obtain a lax bi- $2$ -representation on $B$ by setting $E_{i,j}=E_{2}^{j}F_{1}^{i}$ . We have an injective morphism of differential algebras $\Delta_{E}(B)\to\Delta_{\lambda}(B)$ .

Assume the morphisms (5.2.1) are isomorphisms for all $i$ (this holds for example if $\lambda$ is an isomorphism). Then we have a canonical isomorphism $\Delta_{E}(B)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\Delta_{\lambda}(B)$ . The algebra $\Delta_{\lambda}(B)$ is generated by $B$ and $E_{2}F_{1}$ .

The map $\lambda$ extends (uniquely) to a morphism of algebras $\Delta^{\prime}_{\lambda}(B)\to\Delta_{\lambda}(B)$ that is the identity on $B$ . If $\lambda$ is an isomorphism, then this map is an isomorphism $\Delta^{\prime}_{\lambda}(B)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\Delta_{\lambda}(B)$ .

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2