Generalized Telescope Conjecture
Abstract
We extend Balmer's tensor-triangular geometry to arbitrary presentably symmetric monoidal β-categories π± by introducing the atomic smashing frame. This construction is functorial in π±, recovers the classical Balmer spectrum in the stable compactly-rigidly generated case, and yields a formulation of the telescope conjecture for arbitrary π±. Using dualizable and locally rigid β-categories, we identify smashing ideals with locally rigid localizations and establish recollement results for smashing frames and atomic smashing frames. As a consequence, we obtain a recollement principle for the telescope conjecture. We also establish descent for the telescope conjecture along several natural classes of coverings. As a principal application to chromatic homotopy theory, we analyze the structure of the smashing frame of Sp. We prove a prime decomposition in terms of the smashing frames of the p-local β-categories Sp(p) and, at each prime, an embedding into the product of the smashing frames of the monochromatic layers SpT(n). In particular, the spatiality of the smashing frame of Sp reduces entirely to that of SpT(n). Beyond the stable setting, we characterize the telescope conjecture for β-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 πΌβ-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 β-category.
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BibTeX
@misc{liang2026generalized,
title = {{Generalized Telescope Conjecture}},
author = {Jiacheng Liang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.03375},
eprint = {2609.03375},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03375},
}
RIS
TY - UNPB
TI - Generalized Telescope Conjecture
AU - Jiacheng Liang
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.03375
UR - https://arxiv.org/abs/2609.03375
AB - We extend Balmer's tensor-triangular geometry to arbitrary presentably symmetric monoidal β-categories π± by introducing the atomic smashing frame. This construction is functorial in π±, recovers the classical Balmer spectrum in the stable compactly-rigidly generated case, and yields a formulation of the telescope conjecture for arbitrary π±. Using dualizable and locally rigid β-categories, we identify smashing ideals with locally rigid localizations and establish recollement results for smashing frames and atomic smashing frames. As a consequence, we obtain a recollement principle for the telescope conjecture. We also establish descent for the telescope conjecture along several natural classes of coverings. As a principal application to chromatic homotopy theory, we analyze the structure of the smashing frame of Sp. We prove a prime decomposition in terms of the smashing frames of the p-local β-categories Sp_(p) and, at each prime, an embedding into the product of the smashing frames of the monochromatic layers Sp_T(n). In particular, the spatiality of the smashing frame of Sp reduces entirely to that of Sp_T(n). Beyond the stable setting, we characterize the telescope conjecture for β-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 πΌ_β-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 β-category.
ER -