ScalingStacks

Statements ([008P]) and ([008Q]) follow immediately from corollary 4.4.3 and observation 4.3.5. For statement ([008R]), the inclusion \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z} \hookrightarrow \mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\) induces a symmetric monoidal left adjoint functor \[\begin{aligned} \mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}&\simeq \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}) \xrightarrow{\mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}}\left( \mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\right)} \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\\& \xrightarrow{-\otimes_{\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}} \mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}\mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}) \simeq \mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}. \end{aligned}\] Analogous to proposition 3.4.5, this functor fits into a commuting square in \(\mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\) of the form where the top horizontal morphism is left adjoint to the forgetful functor. Using ([008T]) to consider the bottom horizontal morphism as a morphism in \(\mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})_{\mathrm{add}_{\mathbb{K}}^{B\mathcal Z}/}\), it enhances by proposition 4.1.7 to a symmetric monoidal \(\mathrm{add}_{\mathbb{K}}^{B\mathcal Z}\)-enriched functor \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\rightarrow\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}\). (By commutativity of ([008T]) and observation 4.1.2, we may understand the \(\mathrm{add}_{\mathbb{K}}^{B\mathcal Z}\)-enrichment of \(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}\) as induced by restricting its \(\mathrm{st}_{\mathbb{K}}^{B\mathcal Z}\)-enrichment from observation 4.3.5 along the forgetful functor \(\mathrm{st}_{\mathbb{K}}^{B\mathcal Z}\rightarrow\mathrm{add}_{\mathbb{K}}^{B\mathcal Z}\).) For an algebra \(A\in \mathrm{Alg}(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z})\), it follows from [Lur17, Thm. 4.8.4.6] that \[\mathrm{RMod}_A(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}) \otimes_{\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}} \mathrm{Mod}_{\mathbb{K}}^{\mathcal Z} \simeq \mathrm{RMod}_{A}(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}).\] Thus, the \(\mathrm{add}_{\mathbb{K}}^{B\mathcal Z}\)-enriched functor \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\rightarrow\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}\) restricts to an \(\mathrm{add}_{\mathbb{K}}^{B\mathcal Z}\)-enriched functor between the full subcategories \[\mathrm{Morita}^{\mathrm{cp}}(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}) \rightarrow\mathrm{Morita}^{\mathrm{c}}(\mathrm{Mod}_{\mathbb{K}}^{ \mathcal Z}).\] Its explicit description on additive hom-categories can be unpacked from observation 4.1.2. ◻