Lemma 3.5.7 (005V) · ScalingStacks

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. ◻