We study how smashing Bousfield localizations behave under various equivariant functors. We show that the analogs of the smash product and chromatic convergence theorems for the Real Johnson–Wilson theories \(E_{\mathbb {R}}(n)\) hold only after Borel completion. We establish analogous results for the \(C_{2^n}\)-equivariant Johnson–Wilson theories constructed by Beaudry, Hill, Shi, and Zeng. We show that induced localizations upgrade the available norms for an \(N_\infty \)-algebra, and we determine which new norms appear. Finally, we explore generalizations of our results on smashing localizations in the context of a quasi-Galois extension of \(E_\infty \)-rings.

我们研究了在各种等变函子下，破坏性的布斯菲尔德定位如何表现。我们证明，真正的约翰逊-威尔逊理论的粉碎积和色收敛定理的类比\(E_{\mathbb {R}}(n)\)仅在 Borel 完成后才成立。我们为Beaudry、Hill、Shi 和 Zeng 构建的\(C_{2^n}\)等效约翰逊-威尔逊理论建立了类似的结果。我们证明诱导定位升级了\(N_\infty \)代数的可用范数，并且我们确定出现哪些新范数。最后，我们在\(E_\infty \)环的准伽罗瓦扩展的背景下探索了粉碎本地化结果的概括。