ScalingStacks

Proof.

For the first statement, note that \(\mathcal A\) has enough projective objects (see example 3.6.3) and let \(\mathcal D_-(\mathcal A)\) be the dg-nerve of the differential graded category of bounded-below chain complexes of \(1\)-projective objects (i.e. projective objects in the standard abelian sense). Let \(\mathcal D_{-}(\mathcal A)_{\geq 0}\) be the full subcategory on the chain complexes with vanishing homology in negative degrees. Entirely analogous¹⁴ to the proof of [Lur17, Prop. 1.3.3.14], the Dold-Kan correspondence shows that \(\mathcal D_{-}(\mathcal A)_{\geq 0} \simeq \mathcal P^{\Sigma}(\mathcal A^{\mathrm{c}1\mathrm{p}})\). Since any \(1\)-projectively generated presentable abelian \(1\)-category is Grothendieck abelian  [Lur17, Def. 1.3.5.1], it follows from [Lur17, Prop. 1.3.5.24, Def. 1.3.5.8, Prop. 1.3.5.13] that there is a fully faithful embedding \(\mathcal D_{-}(\mathcal A) \rightarrow\mathcal D(\mathcal A)\) with image the chain complexes with bounded-below homology. In particular, this embedding identifies \(\mathcal D_{-}(\mathcal A)_{\geq 0}\) with \(\mathcal D(\mathcal A)_{\geq 0}\).

For the second statement, since the \(t\)-structure \((\mathcal D(\mathcal A)_{\leq 0}, \mathcal D(\mathcal A)_{\geq 0})\) on \(\mathcal D(\mathcal A)\) is right-complete [Lur17, Prop. 1.3.5.21], it follows that \(\mathcal D(\mathcal A)\) is the stabilization of \(\mathcal D(\mathcal A)_{\geq 0}\); since \(\mathcal D(\mathcal A)_{\geq 0} = \mathcal P^{\Sigma}(\mathcal A^{\mathrm{c}1\mathrm{p}})\) is presentable this stabilization is given by tensoring with \(\mathrm{Sp}\) by [Lur17, Ex. 4.8.1.23]. The equivalence \(\mathcal P^{\Sigma}(\mathcal A^{\mathrm{c}1\mathrm{p}}) \otimes \mathrm{Sp}\simeq \operatorname{Ind}{\mathbf K}^b(\mathcal A^{\mathrm{c}1\mathrm{p}})\) follows then from the definition of \((-)^{\mathrm{fin}}\) in proposition 3.4.5 and its equivalence with \({\mathbf K}^b\) from corollary 3.4.10. ◻

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]). ◻