ScalingStacks

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 \(\mathrm{Mod}_{\mathbb{K}}(\mathrm{Sp}_{\geq 0})\in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}})\).

[005S]

Notation 3.5.4.

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 \(\mathrm{Mod}_{\mathbb{K}}\).

[005T]

Example 3.5.5.

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.

[005U]

Observation 3.5.6.

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 \(\mathrm{Mod}^{\geq 0}_{\mathbb{K}}\).

[005V]

Lemma 3.5.7.

The following hold:

1. 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 \(\mathbb{K}_{\mathbb{K}}\).

2. 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 \(\mathbb{K}_{\mathbb{K}}\).

[005Y]

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 \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\):

[005Z]

Proposition 3.5.8.

For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\) there is a symmetric monoidal equivalence \[{\mathbf K}^b(\mathrm{CProj}_{\mathbb{K}}) \simeq \mathrm{Perf}_{\mathbb{K}}.\]

[0060]

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]). ◻