[006U]
Proposition 3.6.9.
Let \(Z\) be a discrete monoid, \(k\) a discrete commutative ring, and \(A\) a discrete \(Z\)-graded (not necessarily commutative) \(k\)-algebra.
The \(\infty\)-category \(\left(\mathrm{RMod}_{HA}(\mathrm{Mod}_{Hk}^{\geq 0, Z}) \right)^{cp}\) is equivalent to \(\mathrm{grmod}_A^{\mathrm{gr-cp}}\). In particular, it is a \(1\)-category.
The \(\infty\)-category \(\left( \mathrm{RMod}_{HA}(\mathrm{Mod}_{Hk}^Z) \right)^{c}\) is equivalent to the \(\infty\)-category \({\mathbf K}^b(\mathrm{grmod}_A^{\mathrm{gr-cp}})\).
The \(\infty\)-category \(\mathrm{RMod}_{HA}(\mathrm{Mod}_{Hk}^{\geq 0, Z})\) is equivalent to the \(\infty\)-category \(\mathcal D(\mathrm{grmod}_A)_{\geq 0}\).
The \(\infty\)-category \(\mathrm{RMod}_{HA}(\mathrm{Mod}_{Hk}^{Z})\) is equivalent to the (unbounded) derived \(\infty\)-category \(\mathcal D(\mathrm{grmod}_A)\).
[006Z]
Proof.
The \(\infty\)-category \(\mathrm{Mod}_{Hk}^{\geq 0, Z} = \mathrm{Fun}(Z, \mathrm{Mod}_{Hk}^{\geq 0})\) is generated by the set of compact \(1\)-projective objects \(Hk[z]\) for \(z\in Z\), i.e. the ground ring \(k\) in homological degree zero, and grading-degree \(z \in Z\). Hence, by lemma 3.2.12.([004I]), \(\mathrm{RMod}_{HA}(\mathrm{Mod}_{Hk}^{\geq 0, Z})\) is generated by shifted-free modules \(HA[z] = HA \otimes_{Hk} Hk[z]\) for \(z\in Z\). By lemma 3.2.9.([0043]), the compact-projective objects of \(\mathrm{RMod}_{HA}(\mathrm{Mod}_{Hk}^{\geq 0, Z})\) are retracts of finite direct sums of such modules, and hence are precisely the graded-compact-projective modules. This proves ([006V]).
For ([006X]), note that \(\mathrm{RMod}_{HA}(\mathrm{Mod}_{Hk}^{\geq 0, Z})\) is projectively generated (see lemma 3.2.12), and hence equivalent to \[\mathcal P^{\Sigma}\left(\mathrm{RMod}_{HA}(\mathrm{Mod}_{Hk}^{\geq 0, Z})^{\mathrm{cp}}\right) = \mathcal P^{\Sigma}(\mathrm{grmod}_{A}^{\mathrm{gr-cp}}).\] Since \(\mathrm{grmod}_A^{\mathrm{gr-cp}}\) is the full subcategory on the compact 1-projectives in the \(1\)-projectively generated presentable abelian category \(\mathrm{grmod}_A\), it follows from proposition 3.6.6.([006N]) that this is equivalent to \(\mathcal D(\mathrm{grmod}_A)_{\geq 0}\).
Statement ([006Y]) follows from proposition 3.6.6.([006P]) since by [Lur17, Thm. 4.8.4.6], \(\mathrm{RMod}_{HA}(\mathrm{Mod}_{Hk}^{\geq 0, Z}) \otimes \mathrm{Sp}\simeq \mathrm{RMod}_{HA}(\mathrm{Mod}_{Hk}^{\geq 0, Z} \otimes \mathrm{Sp}) \simeq \mathrm{RMod}_{HA} (\mathrm{Mod}_{Hk}^Z).\)
Statement ([006W]) then follows since \(\mathcal D(\mathrm{grmod}_A) \simeq \operatorname{Ind}({\mathbf K}^b(\mathrm{grmod}_A^{\mathrm{gr-cp}}))\) by proposition 3.6.6.([006P]). ◻