ScalingStacks

Example 4.2.10.

As discussed in example 3.5.15, if \(\mathcal Z\) is a discrete (i.e. ordinary) commutative monoid \(Z\), then \((\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z})^{\mathrm{cp}}\) and \((\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z})^c\) are the \(\infty\)-categories of finitely supported functors \(\mathrm{Fun}^{\text{fin.supp.}}(Z, \mathrm{CProj}_{\mathbb{K}})\) and \(\mathrm{Fun}^{\text{fin.supp.}}(Z, \mathrm{Perf}_{\mathbb{K}})\), i.e. of functors that vanish on all but finitely many elements of \(Z\). In particular, in the case of grading by a discrete monoid \(Z\), we obtain the following equivalences:\[\begin{aligned} \mathrm{Pr}^{\mathrm{L},\mathrm{cp}}_{\mathrm{Mod}_{\mathbb{K}}^{\geq 0, Z}} &\simeq \mathrm{Mod}_{\mathrm{Fun}^{\text{fin.supp.}}(Z, \mathrm{CProj}_{\mathbb{K}})}(\mathrm{add}) \\ \mathrm{Pr}^{\mathrm{L},\mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{ Z}} &\simeq \mathrm{Mod}_{\mathrm{Fun}^{\text{fin.supp.}}(Z, \mathrm{Perf}_{\mathbb{K}})}(\mathrm{st}) \end{aligned}\]