Generalized Telescope Conjecture

Abstract: We introduce the atomic smashing frame, extending the Balmer spectrum from tensor-triangular geometry to an arbitrary presentably symmetric monoidal $\infty$-category $\mathcal{V}$. This yields a formulation of the telescope conjecture for $\mathcal{V}$ and recovers the classical Balmer spectrum in the stable compactly-rigidly generated case. Exploiting dualizable and rigid $\infty$-categories, we establish a correspondence between smashing ideals and locally rigid localizations. This leads to a recollement theorem for smashing frames in the (pre)stable setting, together with an atomic refinement in the stable compactly-rigidly generated case. As a major application to chromatic homotopy theory, we show that the natural projections induce an embedding of the smashing frame of $\mathrm{Sp}$ into the product of the smashing frames of the monochromatic layers $\mathrm{Sp}<em>{T(n)}$, over all primes and heights. In particular, the spatiality of the smashing frame of $\mathrm{Sp}$ reduces entirely to that of $\mathrm{Sp}</em>{T(n)}$. In the unstable setting, we characterize the telescope conjecture for $\infty$-topoi in terms of smashing fields, and for connective module categories and hypercomplete connective sheaves in terms of Pierce-type conditions. Finally, we introduce the Serre smashing frame. Over a connective $\mathbb{E}_\infty$-ring $R$, this frame sits between the atomic and usual smashing frames, providing an intermediate structural layer in the study of the telescope conjecture for the connective $R$-module category.