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Math-2262 · mathematics · Topology

Morava K- and E-theory 5 — Find the shadow of the thick subcategory theorem in the K(n)-local category.

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.
Fix a prime $p$ and a positive chromatic height $n\ge1$. In the category of dualizable $K(n)$-local spectra at $p$, classify thick tensor ideals. A thick tensor ideal is closed under cofibers, retracts, and tensoring with any dualizable object. Does their inclusion order give the expected chromatic filtration analogous to the thick-subcategory theorem for finite spectra?

Formulation references: [S1], [S2].

Short English statement

Fix a prime $p$ and a positive chromatic height $n\ge1$. In the category of dualizable $K(n)$-local spectra at $p$, classify thick tensor ideals. A thick tensor ideal is closed under cofibers, retracts, and tensoring with any dualizable object. Does their inclusion order give the expected chromatic filtration analogous to the thick-subcategory theorem for finite spectra?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2262-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2262

Manuscript 1 (family 311, 2026-09-24): The Hovey–Strickland theorem identifies exactly the n+2 predicted thick tensor ideals of all dualizable K(n)-local spectra, for every prime and positive height; it does not confuse dualizable spectra with compact finite-type-n objects.

Morava K- and E-theory 5 — Find the shadow of the thick subcategory theorem in the K(n)-local category.

OpenAI's claim

At every prime and positive height n, stabilizer-invariant primes and radicals are claimed to follow height strata. The tensor-ideal count for dualizable K(n)-local spectra is n+2, with n+1 points forming the Balmer chain. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Fix a prime $p$ and $n\ge1$. Let $\mathcal D_{(n,p)}$ be the category of dualizable $K(n)$-local spectra. For each $0\le k\le n$, choose a finite type-$k$ spectrum $F(k)$ (its first nonzero Morava $K$-theory is height $k$) and let $\mathcal D_k$ be the thick tensor ideal generated by $L_{K(n)}F(k)$. OpenAI claims that all thick tensor ideals are exactly \[ \mathcal D_{(n,p)}=\mathcal D_0\supsetneq\mathcal D_1\supsetneq\cdots\supsetneq\mathcal D_n\supsetneq\mathcal D_{n+1}=0. \] This gives $n+2$ ideals; choosing a different finite type-$k$ generator produces the same ideal. Thick tensor ideals are closed under finite cofibers, retracts and tensoring with dualizable objects.

Outline of the argument

Manuscript 1 (family 311, 2026-09-24): In characteristic p, Frobenius separation converts any orbit-vanishing analytic equation into a Honda endomorphism relation. Fargues--Fontaine bundle geometry and a determinant-degree argument contradict that relation using independent vectors. The characteristic-zero period-map branch handles the other primes, after which minimal-prime analysis and the established support comparison transfer the height chain to the tensor category.

Formal verification

Family 311 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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