ScalingStacks

For \(\mathbb V\in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\), \(A, B \in \mathrm{Alg}(\mathbb V)\) and \({}_{A}M_B \in{}_A\mathrm{BMod}_B(\mathbb V)\), the functor \[- \otimes_{A}M \colon \mathrm{RMod}_{A}(\mathbb V) \rightarrow\mathrm{RMod}_{B}(\mathbb V)\] preserves compact projective objects if and only if \(M_B\), viewed as a right \(B\)-module, is a compact projective object in \(\mathrm{RMod}_{B}(\mathbb V)\). The equivalence from proposition 4.4.1.([008G]) restricts to \[{}_{A}\mathrm{BMod}^{\mathrm{cp}}_{B}(\mathbb V) \simeq \mathrm{Fun}^{L, \mathrm{cp}}_{\mathbb V}(\mathrm{RMod}_A(\mathbb V), \mathrm{RMod}_B(\mathbb V)).\]