ScalingStacks

Definition 4.3. The motivic extension algebra $E$ is the $\mathbb{Z}$-graded locally unital algebra

$$E=\operatorname{End}^{\bullet}_{\operatorname{DM}_{G}(\mathcal{N},\mathbb{Q})}(\mathcal{T}^{Spr})$$

which has a grading induced by the autoequivalence $\langle 1\rangle=(1)[2],$ see (2.3). More explicitly, for $n\in\mathbb{Z}$ the $n$-th graded part of $E$ is

$$E^{n}=\bigoplus_{i,j}\operatorname{Hom}_{\operatorname{DM}_{G}(\mathcal{N},\mathbb{Q})}(\mu_{i,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{i}}),\mu_{j,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{j}})(n)[2n]).$$

For every prime $\ell\neq 0$ we define the $\ell$-adic extension algebra $E^{\acute{e}t}_{\ell}$ in the same way, replacing $\operatorname{DM}_{G}(\mathcal{N},\mathbb{Q})$ by $\operatorname{D}_{G}(\mathcal{N},\mathbb{Q}_{\ell}).$