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Math-3021 · mathematics · Topology

Filtered chromatic splitting at generic primes

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For $n\geq1$ and prime $p>n+1$, does $L_{n-1}L_{K(n)}S_p^\wedge$ admit an ordered $2^n$-stage cofiber filtration indexed by $I\subseteq\{1,\ldots,n\}$, with initial cofiber $L_{n-1}S$ and later cofibers \[ \Sigma^{-\sum_{i\in I}(2i-1)}L_{n-\max I}S, \] first-stage map the canonical localization unit, and one compatible family $y_i$ in degree $1-2i$ whose products give the corresponding $K(t)$-local sphere bases for every $1\leq t<n$? Here all local spheres use the derived $p$-complete sphere.

Short English statement

For every $n\geq1$ and prime $p>n+1$, does $L_{n-1}L_{K(n)}S_p^\wedge$ have an ordered cofiber filtration of $2^n$ stages indexed by $I\subseteq\{1,\ldots,n\}$, starting with cofiber $L_{n-1}S$ and the canonical localization unit as its first-stage map, with later cofibers $\Sigma^{-\sum_{i\in I}(2i-1)}L_{n-\max I}S$, and a single compatible family $y_i$ of degrees $1-2i$ whose products give the corresponding $K(t)$-local sphere bases for all $1\leq t<n$, using the derived $p$-complete sphere for every local sphere?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-3021-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-3021

Manuscript 1 (family 318, 2026-09-25): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Filtered chromatic splitting at generic primes

OpenAI's claim

For n at least one and p greater than n+1, lower-height localization of the K(n)-local sphere is claimed to have an ordered $2^n$-stage filtration with the classical chromatic-splitting cofibers and canonical initial unit. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For $n\ge1$ and prime $p>n+1$, put $S=S_p^\wedge$, $D=L_{K(n)}S$ and $X_{n,p}=L_{n-1}D$, where $L_j=L_{E(j)}$. OpenAI claims a filtration in $E(n-1)$-local $S$-modules with $2^n$ cofibers indexed by subsets $I\subseteq\{1,\ldots,n\}$: \[ C_\varnothing=L_{n-1}S,\qquad C_I=\Sigma^{-d(I)}L_{n-\max I}S\ (I\ne\varnothing),\qquad d(I)=\sum_{i\in I}(2i-1). \] The empty subset is first, the remaining maxima are nondecreasing, and the first-layer map is the canonical unit. It claims a single family $y_i\in\pi_{1-2i}D$ whose ordered products $y_I$ give, for every $1\le t<n$, equivalences \[ L_{K(t)}\left(\bigvee_{I\subseteq\{1,\ldots,n-t\}}\Sigma^{-d(I)}S\xrightarrow{(y_I)}D\right). \] A filtered extension is asserted, not a split wedge decomposition before localization.

Outline of the argument

Manuscript 1 (family 318, 2026-09-25): Finite marking and transfer calculations first establish coefficient finiteness. A separately constructed integral family of sphere-cochain classes has normalized exterior products acting as bases under lower-height tests. Removing blocks in decreasing chromatic height yields the ordered cofibers; independent rational dimension counts supply the final block while retaining the extensions between stages.

Formal verification

Family 318 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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