For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\) there is a symmetric monoidal equivalence \[{\mathbf K}^b(\mathrm{CProj}_{\mathbb{K}}) \simeq \mathrm{Perf}_{\mathbb{K}}.\]
Proof.
Starting with the definition of \({\mathbf K}^b=(-)^{\mathrm{fin}}\) in proposition 3.4.5, we obtain the equivalence \[{\mathbf K}^b(\mathrm{CProj}_{\mathbb{K}}) := \left( \mathcal P^{\Sigma}(\mathrm{CProj}_{\mathbb{K}})\otimes_{\mathrm{Sp}_{\geq 0}}\mathrm{Sp}\right)^{c} \simeq \left( \mathrm{Mod}_{\mathbb{K}}(\mathrm{Sp}_{\geq 0}) \otimes_{\mathrm{Sp}_{\geq 0}}\mathrm{Sp}\right)^{c}\simeq \left( \mathrm{Mod}_{\mathbb{K}}(\mathrm{Sp})\right)^{c} =: \mathrm{Perf}_{\mathbb{K}}\] where the last step follows from proposition 3.1.8.([0036]). ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2