For \(k\) an ordinary commutative ring, we let \(\mathrm{mod}_k\) denote the ordinary symmetric monoidal \(1\)-category of \(k\)-modules.
3.5 \(\infty\)-categories of graded modules[005L]
The categories appearing in this paper will not just be additive or stable, but will typically be enriched in chain complexes of \(k\)-modules for a commutative ring \(k\), equipped with an additional \(\mathbb{Z}\)-grading. In this section, we recall the necessary technical machinery to address this coherently. This machinery will apply more generally to \(\mathbb E_{\infty}\)-ring spectra, i.e. commutative algebra objects in \(\mathrm{Sp}\). In §3.6, we relate these structures with possibly more familiar variants of derived categories. Similar definitions are discussed in [Lur18].
3.5.1 \(\mathbb{K}\)-modules[005M]
We now discuss the \(\infty\)-categorical analog of \(\mathrm{mod}_k\). It follows from lemma 3.2.12.([004J]) that for an \(\mathbb E_{\infty}\)-ring spectrum \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\), the \(\infty\)-category \(\mathrm{Mod}_{\mathbb{K}}(\mathrm{Sp})\) is a compactly generated stable, presentably symmetric monoidal category, i.e. \(\mathrm{Mod}_{\mathbb{K}}(\mathrm{Sp}) \in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}})\).
For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\), we write \(\mathrm{Mod}_{\mathbb{K}}\) for the category of \(\mathbb{K}\)-modules \(\mathrm{Mod}_{\mathbb{K}}(\mathrm{Sp}) \in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}})\) and \(\mathrm{Perf}_{\mathbb{K}}\) for the category of perfect \(\mathbb{K}\)-modules \(\mathrm{Perf}_{\mathbb{K}} \coloneqq \mathrm{Mod}_{\mathbb{K}}(\mathrm{Sp})^{\mathrm{c}} \in \mathrm{CAlg}(\mathrm{st})\).
The symmetric monoidal equivalence \(\operatorname{Ind}\colon \mathrm{st}\rightarrow\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}\) transports \(\mathrm{Perf}_{\mathbb{K}}\) to \(\mathrm{Mod}_{\mathbb{K}}\) and vice versa.
The main application of this paper will only be concerned with the case that \(\mathbb{K}= Hk\) is an Eilenberg-MacLane spectrum of a classical commutative ring \(k\). In this case, \(\mathrm{Mod}_{\mathbb{K}}\) is equivalent to the unbounded derived \(\infty\)-category \(\mathcal D(\mathrm{mod}_k)\) of the abelian category \(\mathrm{mod}_k\) of \(k\)-modules [Lur17, Thm 7.1.2.13] with symmetric monoidal structure given by the derived tensor product \(-\otimes^L_k-\). The \(\infty\)-category \(\mathrm{Perf}_{\mathbb{K}}\) is equivalent to its full subcategory on the perfect chain complexes, i.e. the chain complexes quasi-isomorphic to a bounded complex of finitely generated projective \(k\)-modules.
Since example 3.5.3 is the situation relevant to our paper, the reader can safely view \(\mathbb{K}\) as a classical ring \(k\) and \(\mathrm{Mod}_{\mathbb{K}}\) as \(\mathcal D(\mathrm{mod}_k)\). The situation of example 3.5.3 will be discussed in more detail in §3.6.
3.5.2 Compact-projective \(\mathbb{K}\)-modules[005R]
As above, we will be concerned with the additive variants of the notions in §3.5.1. Let \(\mathbb{K}\) be a connective \(\mathbb E_{\infty}\)-ring spectrum, i.e. a commutative algebra \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\). Since \(\mathrm{Sp}_{\geq 0}\in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}})\), it follows from lemma 3.2.12 that the category \(\mathrm{Mod}_{\mathbb{K}}(\mathrm{Sp}_{\geq 0})\in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}})\).
Fix \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\), we write \(\mathrm{Mod}^{\geq 0}_{\mathbb{K}}\) for the \(\infty\)-category of connective \(\mathbb{K}\)-modules \(\mathrm{Mod}_{\mathbb{K}}(\mathrm{Sp}_{\geq 0}) \in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}})\) and \(\mathrm{CProj}_{\mathbb{K}}\) for the category of compact-projective \(\mathbb{K}\)-modules \(\mathrm{CProj}_{\mathbb{K}} \coloneqq \mathrm{Mod}_{\mathbb{K}}(\mathrm{Sp}_{\geq 0})^{\mathrm{cp}} \in \mathrm{CAlg}(\mathrm{add})\).
Note that \(\mathrm{Mod}^{\geq 0}_{\mathbb{K}}\) is a full subcategory of \(\mathrm{Mod}_{\mathbb{K}}\).
For \(\mathbb{K}= Hk\) an Eilenberg-MacLane spectrum of a classical commutative ring \(k\), the \(\infty\)-category \(\mathrm{Mod}^{\geq 0}_{Hk}\) is equivalent to the full subcategory \(\mathcal D(\mathrm{mod}_k)_{\geq 0}\) of the unbounded derived \(\infty\)-category \(\mathcal D(\mathrm{mod}_k)\) of the ring \(k\) on those chain complexes with homology in non-negative homological degree. It follows from lemma 3.5.7 below that the full subcategory \(\mathrm{CProj}_{Hk}\) is equivalent to the \(1\)-category of finitely generated projective \(k\)-modules in the usual sense (with fully faithful inclusion into \(\mathcal D(\mathrm{mod}_k)_{\geq 0}\) as complexes concentrated in degree zero), see also §3.6.
The symmetric monoidal equivalence \(\operatorname{Ind}\colon \mathrm{st}\rightarrow\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}\) transports \(\mathrm{Perf}_{\mathbb{K}}\) to \(\mathrm{Mod}_{\mathbb{K}}\). Similarly, the symmetric monoidal equivalence \(\mathcal P^{\Sigma}\colon \mathrm{add}\rightarrow\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\) transports \(\mathrm{CProj}_{\mathbb{K}}\) to \(\mathrm{Mod}^{\geq 0}_{\mathbb{K}}\).
The following hold:
Let \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\). The rank one free module \(\mathbb{K}_{\mathbb{K}}\) generates \(\mathrm{CProj}_{\mathbb{K}}\) under retracts and finite direct sums; in particular, every object of \(\mathrm{CProj}_{\mathbb{K}}\) is a retract of a finite coproduct of modules isomorphic to \(\mathbb{K}_{\mathbb{K}}\).
Let \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\). The rank one free module \(\mathbb{K}_{\mathbb{K}}\) generates \(\mathrm{Perf}_{\mathbb{K}}\) under retracts and finite colimits; in particular, every object of \(\mathrm{Perf}_{\mathbb{K}}\) is a retract of an iterated finite colimit of modules isomorphic to \(\mathbb{K}_{\mathbb{K}}\).
Proof.
Immediate from lemma 3.2.9 and the fact that \(\mathrm{Mod}_{\mathbb{K}}\) and \(\mathrm{Mod}^{\geq 0}_{\mathbb{K}}\) are compact and compact projectively generated by \(\mathbb{K}_{\mathbb{K}}\) respectively. ◻
If \(k\) is an ordinary ring, then an object of \(\mathrm{Perf}_{Hk}\) can be represented by a bounded chain complex of finitely generated projective \(k\)-modules. This generalizes to any \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\):
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]). ◻
3.5.3 \(\mathcal Z\)-graded \(\mathbb{K}\)-modules[0061]
Given an ordinary monoid \(Z\) and a commutative ring \(k\), the category \(\mathrm{Fun}(Z, \mathrm{mod}_k)\) of \(Z\)-graded \(k\)-modules admits a convolution monoidal structure, for which the tensor product of \(Z\)-graded modules \((M_z)_{z \in Z}\) and \((N_z)_{z\in Z}\) is given by the \(Z\)-graded module which in degree \(z\in Z\) is \(\oplus_{z_1z_2 = z} M_{z_1} \otimes N_{z_2}\). This construction is a special case of the Day convolution monoidal structure on a functor category [Lur17, § 2.2.6]. Here, we focus on the symmetric monoidal case.
We briefly recall this construction of a symmetric monoidal structure on \(\mathrm{Fun}(J, \mathcal C)\) in the case where \(J\) is a small symmetric monoidal \(\infty\)-category and \(\mathcal C\) is a presentably symmetric monoidal \(\infty\)-category.
For \(J\in \mathrm{Cat}_{\infty}\) and \(\mathcal C\in \mathrm{Pr}^\mathrm{L}\), the functor \(\mathcal C\times J \rightarrow\mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\), \[ (c,j)\mapsto c \otimes \mathrm{Hom}_{J}(-, j) \in \mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\] (where \(\otimes\) denotes the action of \(\mathcal S\) on \(\mathcal C\) inherited from the presentability of \(\mathcal C\)) induces an equivalence \[ \mathcal C\otimes \mathcal P(J) \simeq \mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\] in \(\mathrm{Pr}^\mathrm{L}\) (where \(\otimes\) denotes the tensor product of \(\mathrm{Pr}^\mathrm{L}\)).
Proof.
Consider the chain of equivalences \[\mathcal C\otimes \mathcal P(J) \simeq \mathrm{Fun^L}(\mathcal P(J), \mathcal C^{\mathrm{op}})^{\mathrm{op}} \simeq \mathrm{Fun}(J, \mathcal C^{\mathrm{op}})^{\mathrm{op}} \simeq \mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\] Here, the first equivalence follows from ([002Z]), the second equivalence is the universal property of the Yoneda embedding [Lur09, Thm. 5.1.5.6], and the last records the interplay between functor categories and opposites. Precomposing this equivalence with the inclusion functor \(\mathcal C\times J \rightarrow\mathcal C\otimes \mathcal P(J)\) (which is cocontinuous in its second argument) unpacks to the functor ([0063]). ◻
Assume \(J\in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\) and \(\mathcal C\in \mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\). Since \(\mathcal P\colon \mathrm{Cat}_{\infty}\rightarrow\mathrm{Pr}^\mathrm{L}\) is symmetric monoidal by proposition 3.1.6, it follows that for \(J \in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\), the \(\infty\)-category \(\mathcal P(J)\) inherits a presentably symmetric monoidal structure, i.e. \(\mathcal P(J) \in \mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\). Then ([0064]) provides the following Day convolution monoidal structure on \(\mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\).
Let \(J\in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\) and \(\mathcal C\in \mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\). Then \(\mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\) inherits a presentably symmetric monoidal structure from the tensor product \(\mathcal C\otimes \mathcal P(J)\) of commutative algebras in \(\mathrm{Pr}^\mathrm{L}\).
By [BS24, Prop. 3.10], this construction agrees with the Day convolution structure on functor categories, as e.g. defined in [Lur17, Rem. 2.2.6.8], also see [BS24, Thm. 3.1]. Explicitly, the tensor product of functors \(F\colon J^{\mathrm{op}} \rightarrow\mathcal C\) and \(G \colon J^{\mathrm{op}} \rightarrow\mathcal C\) is given by the left Kan extension of the functor \(J^{\mathrm{op}} \times J^{\mathrm{op}} \xrightarrow{F\otimes G}\mathcal C\) along the tensor product \(J^{\mathrm{op}} \times J^{\mathrm{op}} \rightarrow J^{\mathrm{op}}\).
If \(J \in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\) and \(\mathcal C\) is in \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\) or \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\), then \(\mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\) with its Day convolution monoidal structure is also in \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\) or \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\), respectively.
Proof.
The Day convolution monoidal structure was defined by identifying \(\mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\) with \(\mathcal C\otimes \mathcal P(J)\). The presheaf category \(\mathcal P(J)\) is an object of \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\) (in fact, it is generated by a small set of objects which commute with all small colimits). Hence, if \(\mathcal C\) is in \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\) or in the subcategory \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\), then so is \(\mathcal C\otimes \mathcal P(J)\). ◻
The monoidal unit \(I \in J\) of any symmetric monoidal \(\infty\)-category \(J\in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\) induces a symmetric monoidal functor \(\mathcal S\rightarrow\mathcal P(J)\) left adjoint to the evaluation functor \(\mathrm{ev}_{I} \colon \mathcal P(J) \rightarrow\mathcal S\), and explicitly given by sending a space \(X\) to the functor \(\mathrm{Hom}_{J}(-, I) \times X\colon J^{\mathrm{op}} \rightarrow\mathcal S\). It follows that for any presentably symmetric monoidal category \(\mathcal C\), there is a symmetric monoidal left adjoint \[\mathcal C\simeq \mathcal C\otimes \mathcal S\rightarrow\mathcal C\otimes\mathcal P(J) \simeq \mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\] to the evaluation functor \(\mathrm{ev}_{I}\colon\mathrm{Fun}(J^{\mathrm{op}}, \mathcal C) \rightarrow\mathcal C\), explicitly given by sending \(c\in \mathcal C\) to the functor \(\mathrm{Hom}_{J}(-, I) \otimes c \colon J^{\mathrm{op}} \rightarrow\mathcal C\).
We will particularly focus on gradings by a homotopy coherent abelian monoid, i.e. a \(\mathcal Z\in \mathrm{CAlg}(\mathcal S)\).
Let \(\mathcal Z\in \mathrm{CAlg}(\mathcal S)\) and recall Day convolution from corollary 3.5.10.
For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\), we define the \(\infty\)-category of \(\mathcal Z\)-graded connective \(\mathbb{K}\)-modules \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z} \in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\) as the functor category \(\mathrm{Fun}(\mathcal Z, \mathrm{Mod}^{\geq 0}_{\mathbb{K}})\) with the Day convolution structure.
For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\), we define the \(\infty\)-category of \(\mathcal Z\)-graded \(\mathbb{K}\)-modules \(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z} \in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\) to be the functor category \(\mathrm{Fun}(\mathcal Z, \mathrm{Mod}_{\mathbb{K}})\) with the Day convolution structure.
Following example 3.5.3, if \(\mathcal Z\) is a discrete (i.e. ordinary) commutative monoid \(Z\) and \(\mathbb{K}= Hk\) the Eilenberg-MacLane spectrum of an ordinary commutative ring \(k\), the \(\infty\)-category \(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\) is the unbounded derived \(\infty\)-category \(\mathcal D(\mathrm{mod}_k^Z)\) of the ordinary abelian \(1\)-category \(\mathrm{mod}_k^{Z}\coloneqq \mathrm{Fun}(Z, \mathrm{mod}_k)\) of \(Z\)-graded \(k\)-modules. This will be discussed in more detail in subsection 3.6.
Unpacking Day convolution from corollary 3.5.10 in these terms, the tensor product of an ordinary \(k\)-module \(M\) concentrated in degree \(z\in Z\) and an ordinary \(k\)-module \(N\) concentrated in degree \(w\in Z\) is given by the derived tensor product \(M\otimes_k^L N\) concentrated in degree \(z+w\in Z\).
Still in the setup of example 3.5.15, the \(\infty\)-categories \(\left(\mathrm{Mod}_{Hk}^{\geq 0, Z}\right)^{\mathrm{cp}}\) and \(\left(\mathrm{Mod}_{Hk}^{Z}\right)^{\mathrm{c}}\) may be identified with the full subcategories \(\mathrm{Fun}^{\mathrm{fin.supp.}}(Z, \mathrm{CProj}_{k})\) and \(\mathrm{Fun}^{\mathrm{fin.supp.}}(Z, \mathrm{Perf}_{k})\) of the functor \(\infty\)-categories \(\mathrm{Fun}(Z, \mathrm{CProj}_{k})\) and \(\mathrm{Fun}(Z, \mathrm{Perf}_{k})\), respectively, on the finitely supported functors, i.e. functors that vanish on all but finitely many elements of \(Z\).
Assume \(\mathcal Z\in \mathrm{CAlg}(\mathcal S)\) and \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\). The symmetric monoidal functor \(- \otimes \mathrm{Sp}\colon \mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\rightarrow\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}\) from construction 3.4.4 takes \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}\) with its Day convolution monoidal structure to \(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\) with its Day convolution monoidal structure. Indeed, we have the following sequence\[\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z} \otimes \mathrm{Sp}\simeq \mathcal P(\mathcal Z) \otimes \mathrm{Mod}_{\mathbb{K}}^{\geq 0} \otimes \mathrm{Sp}\simeq \mathcal P(\mathcal Z) \otimes \mathrm{Mod}_{\mathbb{K}} \simeq \mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\] of symmetric monoidal equivalences. In particular, it follows that the fully faithful inclusion \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z} \hookrightarrow \mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\) is symmetric monoidal and hence a morphism in \(\mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\).
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2