ScalingStacks

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.