The strong chromatic splitting conjecture

About 7 years old · traced to 2019

Let $E_{n-1}$ -local spectra be localized with respect to Morava $E$ -theory, let $S_p^0$ be the $p$ -complete sphere spectrum, and let

\Lambda_R(X_1,\ldots,X_n)

denote the exterior-algebra-type wedge of $R$ and all smash products of distinct listed spectra over $R$ . Strong chromatic splitting conjecture. If $p\neq2$ , or if $p=2$ and $n$ is odd, then there is an equivalence

L_{n-1}L_{K(n)}S^0\simeq\Lambda_{L_{n-1}S_p^0}(L_{n-i}S^{1-2i}:1\leq i\leq n)

in the category of $E_{n-1}$ -local spectra, with $\iota$ corresponding to the unit. If $p=2$ and $n$ is even, then there is an $E_{n-1}$ -local equivalence

L_{n-1}L_{K(n)}S^0\simeq\Lambda_{L_{n-1}S_2^0}(L_{n-i}S^{1-2i}:1\leq i\leq n)\wedge\Lambda_{L_{n-1}S_2^0}(L_{n-1}S^{-2}/2),

with $\iota\wedge\iota$ corresponding to the unit. This is a proposed revision of the original conjecture, incorporating the additional class at $p=n=2$ . It implies the weak chromatic splitting conjecture; the source notes that the weak form holds for finite spectra but fails for the $p$ -completion of $BP$ .

References

Primary source

Tobias Barthel and Agnès Beaudry, “Chromatic structures in stable homotopy theory”, arXiv:1901.09004 (2019).

chromatic-homotopy-theory stable-homotopy-theory

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RemarkAI-assistedClaimed by OpenAI. Claims an ordered 2^n-stage chromatic-overlap filtration for every height n>=1 and prime p>n+1 with the classical cofibers and canonical localization unit as first-stage map. The result is a filtration, not a wedge splitting.See full solutionHide full solution

Claimed by OpenAI. Claims an ordered 2^n-stage chromatic-overlap filtration for every height n>=1 and prime p>n+1 with the classical cofibers and canonical localization unit as first-stage map. The result is a filtration, not a wedge splitting.

Scope relative to this problem: Related structural progress: ordered 2^n-stage chromatic-overlap filtrations for height n>=1 and primes p>n+1, with classical cofibers and the canonical localization unit as the first map. A filtration does not assert the target exterior-algebra-type wedge splitting or a homotopy retraction.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Filtered-chromatic-splitting-at-generic-primes-September-25-2026/paper.pdf

OpenAI-318-01-Filtered-chromatic-splitting-at-generic-primes.pdf969,082 bytesOpen

RemarkAI-assistedClaimed by OpenAI. Claims explicit eight-stage height-three chromatic-overlap filtrations at every prime at least five, identifying attachments and retaining the canonical first-stage unit; a companion result implies a nonzero first height-one attachment.See full solutionHide full solution

Claimed by OpenAI. Claims explicit eight-stage height-three chromatic-overlap filtrations at every prime at least five, identifying attachments and retaining the canonical first-stage unit; a companion result implies a nonzero first height-one attachment.

Scope relative to this problem: Related height 3 structural progress at primes p>=5: an explicit eight-stage overlap filtration with identified attachments and canonical first-stage unit. The source retains nonzero attachments, including its companion-derived first height 1 attachment. This is not a wedge decomposition or a proof of strong splitting.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-height-three-chromatic-overlap-an-explicit-filtration-and-its-attachments-September-27-2026/paper.pdf

OpenAI-318-02-The-height-three-chromatic-overlap-an-explicit-filtration-and-its-attachments.pdf1,033,537 bytesOpen

RemarkAI-assistedClaimed by OpenAI. Claims that a specified canonical rationalized height-three-to-height-two overlap map is nonzero in degree minus three for primes at least five, contradicting the strong height-three chromatic-splitting formula even as an underlying-spectrum equivalence.See full solutionHide full solution

Claimed by OpenAI. Claims that a specified canonical rationalized height-three-to-height-two overlap map is nonzero in degree minus three for primes at least five, contradicting the strong height-three chromatic-splitting formula even as an underlying-spectrum equivalence.

Scope relative to this problem: The manuscript claims a specified rationalized height 3-to-height 2 overlap map is nonzero in degree -3 at primes p>=5 and thereby contradicts the strong height 3 formula even as an underlying-spectrum equivalence. Retain this exact height, prime range and map; do not infer arbitrary-height or weak-map nonretraction from this companion.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/paper.pdf

OpenAI-318-03-A-rational-obstruction-to-strong-chromatic-splitting-at-height-three.pdf653,648 bytesOpen

RemarkAI-assistedClaimed by OpenAI. Claims failure of finite assembly of the height-three chromatic overlap at prime three from rational, height-one and height-two local spheres by finite sums, shifts, cofibers and retracts in the specified E(2)-local module category.See full solutionHide full solution

Claimed by OpenAI. Claims failure of finite assembly of the height-three chromatic overlap at prime three from rational, height-one and height-two local spheres by finite sums, shifts, cofibers and retracts in the specified E(2)-local module category.

Scope relative to this problem: Related restricted-category obstruction at height 3, prime 3: no finite assembly from rational, height 1 and height 2 local spheres by finite sums, shifts, cofibers and retracts inside the specified E2-local MODULE category. It is not independently attributed as an obstruction to every underlying-spectrum wedge equivalence or canonical weak splitting-map retraction.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Failure-of-finite-assembly-for-a-chromatic-overlap-at-the-prime-three-September-25-2026/paper.pdf

OpenAI-318-04-Failure-of-finite-assembly-for-a-chromatic-overlap-at-the-prime-three.pdf776,545 bytesOpen