---
title: "Generalized Telescope Conjecture"
authors: ["Jiacheng Liang"]
year: 2026
venue: "arXiv:math.AT"
arxiv: "2609.03375"
doi: null
source: "https://arxiv.org/abs/2609.03375"
topics: ["curiosity", "maths-foundations"]
seed_rank: 1584
seed_batch: "curiosity-2026-09-05"
reviewed: "2026-09-05"
pool: "maths-foundations"
relevance_score: 10
lineage: chromatic-homotopy
cites:
  - title: "Generalized Telescope Conjecture"
    url: "https://arxiv.org/abs/2609.03375"
    year: 2026
    arxiv: "2609.03375"
    doi: null
see: []
---

# Generalized Telescope Conjecture

## One-sentence takeaway

Telescope conjecture + Balmer spectrum lifted from tt-geometry to presentably monoidal ∞-categories via atomic smashing frames.

## Why it's lovely

Why you might love this: chromatic homotopy’s signature conjecture, rewritten so Balmer’s spectrum makes sense far beyond compactly generated stable categories. Liang introduces the atomic smashing frame, pushes the telescope conjecture to an arbitrary presentably symmetric monoidal ∞-category V, and recovers classical Balmer when V is stable and compactly-rigidly generated. The smashing-ideal / locally-rigid-localization dictionary is the machine.

## Key ideas

- Atomic smashing frame: Balmer spectrum extended from tensor-triangular geometry to presentably symmetric monoidal ∞-categories V.
- Telescope conjecture formulated for general V; classical Balmer spectrum recovered in the stable compactly-rigidly generated case.
- Correspondence between smashing ideals and locally rigid localizations, via dualizable / rigid ∞-categories.
- Recollement for smashing frames in the (pre)stable setting; atomic refinement when compactly-rigidly generated.
- Chromatic application: the smashing frame of Sp embeds into the product of frames of monochromatic layers Sp_{T(n)}; spatiality of Sp reduces to the T(n) layers.
- Unstable side: telescope conjecture for ∞-topoi via smashing fields; a Serre smashing frame interpolates atomic vs usual frames over connective E∞-rings.

## Caveats

A generalization of the *formulation* and of several structural reductions — not a proof of Ravenel’s classical telescope conjecture. Heavy ∞-categorical machinery; “spatiality reduces to T(n)” is a reduction, not a computation of those frames.

## Implications for Broadside

Chromatic-homotopy sparkle. No engine ticket, but the move (take a spectrum that only made sense for compact objects, rewrite it for presentable monoidal ∞-categories) is the same shape as promoting a type-theoretic slogan to the wild setting where it might fail.

## Links

- arXiv abs: https://arxiv.org/abs/2609.03375
- PDF: https://arxiv.org/pdf/2609.03375.pdf