ScalingStacks

Proposition 3.11. The heart of the weight structure on $\operatorname{DTM}_{G}(k,\mathbb{Q})$ is generated by the objects $\operatorname{Ind}^{G}_{G^{0}}(\mathbb{Q})(n)[2n]$ with respect to isomorphism, direct summands and finite direct sums

$$\operatorname{DTM}_{G}(k,\mathbb{Q})^{w=0}=\langle\operatorname{Ind}^{G}_{G^{0}}(\mathbb{Q})(n)[2n]\mid n\in\mathbb{Z}\rangle_{\cong,\inplus,\oplus}\subset\operatorname{DTM}_{G}(k,\mathbb{Q}).$$