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Telescope conjecture via homological residue fields with applications to schemes

Michal Hrbek
Vol. 7 (2025), No. 1, 199–228
DOI: 10.2140/tunis.2025.7.199
Abstract

For a big tt-category, we give a characterization of the telescope conjecture (TC) in terms of definable ⊗-ideals generated by homological residue fields. We formulate a stalk-locality property of (TC) and prove that it holds in the case of the derived category of a quasicompact quasiseparated scheme, strengthening the 2024 result from Hrbek, Hu and Zhu. As an application, we find strong links between (TC) and separation properties of the adic topology on local rings. This allows us to recover known examples and counterexamples of when (TC) holds over a scheme, as well as to construct some new ones.

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Keywords
tensor triangulated category, telescope conjecture, homological residue field, derived category, scheme
Mathematical Subject Classification
Primary: 18G80, 18N55
Secondary: 13B35, 13H99
Milestones
Received: 26 November 2023
Revised: 6 September 2024
Accepted: 28 September 2024
Published: 7 March 2025
Authors
Michal Hrbek
Institute of Mathematics
Czech Academy of Sciences
Prague
Czech Republic
© 2025 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers).