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Math-2261 · mathematics · Topology

Morava K- and E-theory 1 — Show that pi_* L_K(n) S^0 is finitely generated over the p-adics in each degree.

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For every prime $p$, positive chromatic height $n\ge1$, and integer degree $k$, must $\pi_kL_{K(n)}S^0$ be a finitely generated module over the $p$-adic integers? Here $L_{K(n)}$ is localization of the sphere spectrum at the height-$n$ Morava $K$-theory.

Formulation references: [S1], [S2].

Short English statement

For every prime $p$, positive chromatic height $n\ge1$, and integer degree $k$, must $\pi_kL_{K(n)}S^0$ be a finitely generated module over the $p$-adic integers? Here $L_{K(n)}$ is localization of the sphere spectrum at the height-$n$ Morava $K$-theory.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2261-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2261

Manuscript 1 (family 313, 2026-09-24): Theorem 1.1 is exactly integer-degree finite generation over the p-adic integers for every prime and positive height, and explicitly excludes the different Picard-graded extension mentioned in the background.

Morava K- and E-theory 1 — Show that pi_* L_K(n) S^0 is finitely generated over the p-adics in each degree.

OpenAI's claim

At every prime and positive height, each integer-degree homotopy group of the K(n)-local sphere is claimed to be finitely generated over Z_p, equivalently for every localized finite p-local spectrum. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
At every prime $p$, positive height $n$ and integer degree $t$, OpenAI claims \[ \pi_tL_{K(n)}S_p^\wedge\text{ is finitely generated over }\mathbb Z_p. \] Equivalently it claims the same for $\pi_tL_{K(n)}F$ for every finite $p$-local spectrum $F$. Here $K(n)$ is Morava $K$-theory, $L_{K(n)}$ its localization and $\mathbb Z_p$ the $p$-adic integers. Negative as well as nonnegative degrees are included.

Outline of the argument

Manuscript 1 (family 313, 2026-09-24): Finite division-point markings and compact-support geometry control stabilized transfer images in continuous Morava coefficient cohomology. Laurent-series models and Frobenius contraction remove a positive-order acyclic kernel, while Koszul extension arguments promote stable-image finiteness to the required coefficient groups. Sphere descent completes the module-finiteness conclusion, and finite cofiber sequences and retracts extend it to finite spectra.

Formal verification

Family 313 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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