Title: Generalized Telescope Conjecture
Abstract: We introduce the atomic smashing frame, extending the Balmer spectrum to arbitrary presentably symmetric monoidal ∞-categories and giving a general formulation of the telescope conjecture. We identify smashing ideals with locally rigid localizations and establish recollement and descent results for smashing frames. As a principal application, we study the structure of the smashing frame of Sp. We prove a prime decomposition in terms of the p-local categories Sp_(p), and at each prime construct an embedding into the product of the smashing frames of the monochromatic categories Sp_T(n). In particular, the spatiality problem for the smashing frame of Sp reduces entirely to Sp_T(n). We also discuss connective variants, where the telescope conjecture is governed by algebraic conditions on the heart.
David Lee
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