Corollary 4.4.3 (008I) · ScalingStacks

The following hold.

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

Proof.

We will prove statement ([008J]), the proof of statement ([008K]) is completely analogous. Since \(\mathrm{RMod}_A(\mathbb V)\) is projectively generated by free modules \(v\otimes A\), see lemma 3.2.12.([004E]), where \(v \in \mathbb V\) is compact projective, it suffices to show that \(-\otimes_A M \colon \mathrm{RMod}_{A}(\mathbb V) \rightarrow\mathrm{RMod}_B(\mathbb V)\) preserves compact projective objects if and only if it sends such free modules \(v\otimes A\) to compact projectives in \(\mathrm{RMod}_B(\mathbb V)\) for all compacts \(v\in \mathbb V\). Since the action functor \(\mathbb V\times \mathrm{RMod}_B(\mathbb V) \rightarrow\mathrm{RMod}_B(\mathbb V)\) takes pairs of compact projectives to compact projectives by lemma 3.2.12.([004D]), this in turn is equivalent to the assertion that \(A\otimes_A M_B \simeq M_B \in \mathrm{RMod}_B(\mathbb V)\) is compact projective. ◻