The composite \(\mathrm{add}_{k}^{B\mathbb{Z}}\)-enriched functor \(\mathrm{Sbim}\rightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\) and its further composite \(\mathrm{Sbim}\rightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\hookrightarrow \mathrm{st}^{B\mathbb{Z}}_{k}\) are faithful.
Proof.
We show that the former functor is faithful, the latter functor is a composite with a fully faithful functor and hence also faithful. By construction, the former functor factors as \(\mathrm{Sbim}\rightarrow\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\rightarrow\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\), the first of which is faithful by corollary 6.2.7, and the second is faithful since the induced map on hom-categories between two polynomial algebras \(A\) and \(B\) is given by the full inclusion. \[{}_{A}\mathrm{grbmod}^{\mathrm{gr-cp}}_{B} \hookrightarrow \mathcal D({}_{A} \mathrm{grbmod}_{B})^{\mathrm{gr-perf}}\] by corollary 4.5.8.([0095]). ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2