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Counterexamples to finite generation at chromatic height two

Refutes the Hahn–Wilson conjecture at chromatic height two. For every sufficiently large prime p, constructs a connective p-complete spectrum of exact fp-type two that cannot be built from completed

\mathrm{BP}\langle2\rangle

by finitely many sums, shifts, cones and retracts. The examples nevertheless satisfy the finite and telescopic localization comparisons.

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Counterexamples to the Hahn-Wilson conjecture at height two

Sep 26, 2026·paper.pdf

We disprove the Hahn–Wilson conjecture at height two. For every sufficiently large prime p, we construct a connective p-complete spectrum X of exact fp-type two outside the ordinary thick subcategory generated by the specified standard form of

\mathrm{BP}\langle2\rangle_p^\wedge

. The same spectrum satisfies both localization comparisons

L_2^fX\simeq L_2X

and

L_{T(2)}X\simeq L_{K(2)}X

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