We let \(\mathrm{grmod}_A \coloneqq \mathrm{RMod}_A(\mathrm{mod}_{k}^Z)\) denote the ordinary \(1\)-category of \(Z\)-graded right \(A\)-modules.
3.6.2 Derived \(\infty\)-categories of graded modules[006R]
We return to the main goal of this subsection to give a homological perspective on the constructions of the last sections. Let \(\mathbb{K}\) be a discrete commutative ring \(k\) and \(\mathcal Z\) a discrete commutative monoid \(Z\). Recall the notation \(\mathrm{mod}_{k}^Z\) for the ordinary category of \(Z\)-graded \(k\)-modules. Throughout this subsection, we also fix an ordinary (not necessarily commutative) \(Z\)-graded \(k\)-algebra \(A \in \mathrm{Alg}(\mathrm{mod}_{k}^Z)\).
This category \(\mathrm{grmod}_A\) is a \(1\)-projectively generated, in the sense of definition 3.6.1, presentable abelian \(1\)-category. A standard computation shows that its compact \(1\)-projective objects (i.e. its compact projective objects in the usual abelian sense) are precisely given by the graded-compact projective modules, defined as follows.
An (ordinary) \(Z\)-graded \(A\)-module \(M \in \mathrm{grmod}_A\) is graded-compact-projective if it is a retract of a finite direct sums of grading shifts of the free module \(A\). Let \(\mathrm{grmod}_A^{\mathrm{gr-cp}} \subset \mathrm{grmod}_A\) denote the full subcategory on the graded-compact-projective \(A\)-modules.
In the notation of definition 3.6.1, \(\mathrm{grmod}_A^{\mathrm{gr-cp}} = \left(\mathrm{grmod}_A\right)^{\mathrm{c}1\mathrm{p}}\).
Using proposition 3.6.6, we can identify the \(\infty\)-category \(\mathrm{RMod}_{HA}(\mathrm{Mod}_{Hk}^{\geq 0, Z})\) as well as its various subcategories in terms of homological algebra:
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 \(\mathcal D(\mathrm{grmod}_A)\).
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]). ◻
Motivated by proposition 3.6.9, we call the objects in the full subcategory \(\mathcal D(\mathrm{grmod}_A)^{\mathrm{c}} \subseteq \mathcal D(\mathrm{grmod}_A)\) graded-perfect.
Since \(\mathcal D(\mathrm{grmod}_A)^{\mathrm{c}} \simeq {\mathbf K}^b(\mathrm{grmod}_A^{\mathrm{gr-cp}})\) an object is graded-perfect if it is quasi-isomorphic to a bounded (in either direction) chain complex of graded-compact-projective \(A\)-modules.
We write \(\mathcal D(\mathrm{grmod}_A)^{\mathrm{gr-perf}}\coloneqq \mathcal D(\mathrm{grmod}_A)^{\mathrm{c}}\) for the full subcategory of \(\mathcal D(\mathrm{grmod}_A)\) on the graded-perfect modules.
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2