ScalingStacks

Recall from observation 4.5.10 that \(\mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{\mathbb{Z}})\) has a symmetric monoidal fully faithful \(\mathrm{st}^{B\mathbb{Z}}_{k}\)-enriched functor \[\mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{\mathbb{Z}})\hookrightarrow \mathrm{st}^{B\mathbb{Z}}_{k}.\] We will abuse notation and also denote by \(H_{\mathrm{loc}}\) the monoidal \(\mathrm{st}^{B\mathbb{Z}}_{k}\)-enriched composite \[ H_{\mathrm{loc}}\colon {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\hookrightarrow \mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{\mathbb{Z}})\hookrightarrow \mathrm{st}^{B\mathbb{Z}}_{k}.\] The composite sends a graded \(k\)-algebra \(A\) to the stable \(\infty\)-category of bounded chain complexes of graded-compact-projective right \(A\)-modules, with \(\mathbb{Z}\)-action given by the internal (i.e. non-homological) grading shift.