Local finiteness does not guarantee a finite global construction

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Family 319 · Topology
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-22

1. The question

Some infinite objects can still be built from a standard object using only finitely many operations. A finite construction can therefore be a useful notion of simplicity even when the underlying object has infinitely many nonzero homotopy groups.

The Hahn–Wilson conjecture proposes such a classification for spectra. Roughly, finite presentation of mod-$p$ cohomology, combined with a chromatic height bound, should say exactly when a spectrum can be assembled from a truncated Brown–Peterson spectrum by finite cofibers and retracts.

Family 319 claims counterexamples at height two for every sufficiently large prime. Their local chromatic behavior remains well controlled, including the two relevant telescope comparisons. The asserted obstruction lies in the way the local pieces are joined into a global object.

2. Background and essential definitions

A spectrum is a stable object in which suspension has an inverse. It is connective if its homotopy groups vanish in negative degrees, and bounded below if they vanish below some fixed degree. Derived $p$-completion retains compatible information modulo all powers of a prime $p$.

The Steenrod algebra consists of stable operations on mod-$p$ cohomology. A module over it is finitely presented if finitely many generators and finitely many relations describe the module, even though the module itself may be infinite-dimensional over $\mathbb F_p$.

Following Mahowald–Rezk, a bounded-below $p$-complete spectrum $X$ is fp when $H^*(X;\mathbb F_p)$ is finitely presented over the Steenrod algebra. An equivalent test asks for a nonzero finite spectrum $V$ such that $\pi_*(V\wedge X)$ is finite in total: only finitely many degrees are nonzero, and each group is finite.

Morava $K$-theories $K(j)$ detect chromatic heights, with $K(0)=H\mathbb Q$. A nonzero finite spectrum has type $j$ when $j$ is its first nonvanishing Morava height. An fp-spectrum has fp-type at most $n$ if the total-finiteness test works with a finite spectrum of type $n+1$. Exact fp-type two means at most two but not at most one.

The standard completed truncated Brown–Peterson spectrum has coefficients

$$\pi_*BP\langle2\rangle_p^{\wedge}=\mathbb Z_p[v_1,v_2], \qquad |v_i|=2(p^i-1).$$

The variables record the first two chromatic periodicities. The ordinary thick subcategory generated by this spectrum permits equivalences, finite sums, all integer shifts, cofiber sequences, and retracts. It does not add closure under smashing with arbitrary other spectra.

Write $L_2$ for localization retaining heights zero through two, and $L_2^f$ for the corresponding finite localization. The telescope $T(2)$ is built by repeatedly applying a periodic self-map to a finite type-two spectrum. Agreement of $T(2)$-localization with $K(2)$-localization is a separate condition, not part of the definition of fp-type.

3. Development history and previous partial results

The Hopkins–Smith theorem classifies thick subcategories of finite spectra by chromatic type. The Hahn–Wilson proposal asks for an analogous finite-construction description inside a controlled class of infinite spectra, using truncated Brown–Peterson spectra as generators.

Mahowald–Rezk developed the fp framework and its duality theory. Hahn–Wilson’s redshift theorem supplied further examples through algebraic $K$-theory of suitable truncated Brown–Peterson forms. The very low fp-type cases are classical, and Lee–Pstrągowski proved the global generation conjecture at height one.

Lee–Pstrągowski also established a positive local theorem at every height. The locally fp $K(n)$-local spectra are exactly the ordinary thick closure of Morava $E$-theory $E_n$, equivalently of the localized truncated Brown–Peterson spectrum. The local finiteness test is degreewise after smashing with finite type-$n$ spectra; it differs from the global test requiring finiteness in total.

A natural hope was that this local classification, together with good comparison between finite and ordinary chromatic localization, would imply global generation. M1 claims that this hope fails at height two.

The unrestricted telescope conjecture had already been disproved by Burklund, Hahn, Levy, and Schlank at heights at least two. Those examples do not supply the present counterexample: the required finite-smash homotopy is bounded but not finite in total, so the relevant fp condition is missing. M1 instead keeps the telescope comparisons true for its own example.

The manuscript’s history also distinguishes the tensor-ideal classification discussed in family 311. That classification allows multiplication by other dualizable local objects, whereas the Hahn–Wilson problem fixes a single global generator and ordinary thick closure.

4. What the new manuscripts claim

M1, dated 26 September 2026, states that for every sufficiently large prime $p$, there is a connective $p$-complete spectrum $X$ of exact fp-type two such that

$$X\notin\operatorname{Thick}(BP\langle2\rangle_p^{\wedge}).$$

The same spectrum satisfies

$$L_2^fX\simeq L_2X, \qquad L_{T(2)}X\simeq L_{K(2)}X.$$

Thus the counterexample persists even with these localization comparisons imposed as additional hypotheses. It disproves the universally quantified global generation conjecture at height two.

The paper works with a specified standard form of $BP\langle2\rangle_p^{\wedge}$, obtained as a regular Brown–Peterson module quotient. It invokes the cited uniqueness results for the underlying standard quotient spectrum at the odd primes in question. This is not a claim about every possible unrelated object sharing the displayed coefficient ring.

Its local object $L_{K(2)}X$ still belongs to $\operatorname{Thick}(E_2)$ and has the associated local finiteness and duality properties. Those positive conclusions are compatible with failure of the global construction.

“Sufficiently large” is the theorem’s prime range. One algebraic comparison used in the argument is available for $p>8$, but the operator construction requires larger primes. The comparison’s numerical bound should not be advertised as the final counterexample threshold.

5. Why the advance matters

Importance: 5/5, provisionally. If correct, the result refutes a central proposed classification of chromatically finite infinite spectra and identifies a specific limitation of passing from local descriptions to global constructions.

It does more than exploit bad behavior of telescope localization. The example satisfies the relevant comparisons, so those hypotheses cannot repair the conjecture by themselves. Any replacement must also control the gluing between heights.

The coexistence of local finite generation and global failure is broadly instructive. Each local part can belong to the expected finite-construction class while the map identifying the parts over their overlap introduces an obstruction that no finite global tower can remove.

6. The main idea

Move to an algebraic setting while retaining maps

At large primes, chromatic algebraicity compares the relevant homotopy category with a periodic derived category of sheaves on the moduli of formal group laws. A derived category retains complexes and their maps up to the equivalences needed for homological calculations.

The manuscript uses a homology-compatible triangulated comparison, so finite triangles and retracts can be transported. It first performs completion, inverse limits, and gluing in the algebraic category. It does not assume a mere equivalence of underlying categories automatically preserves every construction being used.

On a reduced height-at-most-two stack, a parameter $t=v_1$ cuts out the height-two locus. Inverting it gives an open part, while completion gives a neighborhood of the closed part. The overlap is described by Laurent-type coefficients, allowing both positive and negative powers.

Choose gluing with controlled but numerous defects

The construction starts with the cone of an operation on the completed standard generator, together with shifts through one period. Call this completed object $Y$. On the overlap, the operation has many independent kernel and cokernel classes. A triangular matching of coefficients controls both defects and keeps its image closed.

The paper pairs kernel and cokernel directions inside a discrete cocompact vector subspace. Because the open part is simple, this subspace can be realized by an open periodic complex $V$ mapping into the overlap $LY$, where $L$ means inversion of $t$.

The global object is the homotopy pullback

$$U=Y\mathbin{\times}^{h}_{LY}V.$$

The complete part contributes a compact open subspace of overlap homotopy. The discrete part complements it up to finite error. Consequently, the combined map has finite kernel and cokernel in each degree, and the pullback has finite homotopy groups degree by degree. Its completion recovers $Y$, which has the desired finite construction locally.

Make every finite global tower impossible

The obstruction compares two growth rates at Laurent precision $N$. A separate operator estimate bounds certain joint kernels by $O(N^{1/3})$. The deliberately paired defects grow at least like $cN^\beta$ for a fixed $\beta>1/3$, with $c>0$.

This discrepancy constrains maps from $U$ to any finite construction from the global generator. After a flat coefficient extension, a map vanishing on homology must also vanish on the homology of the module dual. The dual condition removes the free part of a potential construction stage; the homology condition then passes the obstruction to the preceding stage.

Backward induction rules out every finite tower and its retracts. This is stronger than showing that one chosen presentation fails or that an object cannot be made with a small number of cofibers.

Finally, the algebraic object is transferred back to spectra. Only after obtaining a suitable localized object with degreewise finite homotopy does the paper take its connective cover. A separate transfer argument proves the total-finiteness test required for fp-type. Additional arguments establish exact type two and both localization comparisons.

7. Scope, limitations, and open questions

The result is existential at sufficiently large primes. It does not settle the global conjecture separately at every small prime or provide a smallest counterexample.

The generator and permitted operations matter. Ordinary thick closure is not a tensor ideal, and the global fp classification is not the local classification of dualizable objects. The positive local theorem remains part of the description of the example.

Degreewise finiteness is weaker than finiteness in total. The construction needs an additional argument after taking a connective cover to reach the latter fp test; the two notions are not interchangeable.

Taking connective covers of arbitrary local generators also fails as a shortcut. M1 points out that the connective cover of height-two Morava $E$-theory has an infinite power-series coefficient ring modulo $p$ in degree zero, unlike objects in the claimed global thick closure. This simpler obstruction illustrates why a local finite tower cannot automatically be converted to the desired global one.

A revised classification would need conditions controlling the overlap and its maps, beyond the separate local pieces and telescope comparisons. The manuscript gives the obstruction, rather than proposing a complete replacement theorem.

8. Sources and further reading

M1 supplies the counterexample and growth-based gluing obstruction. Primary context includes Lee–Pstrągowski, The monochromatic Hahn–Wilson conjecture; Mahowald–Rezk’s fp-spectrum duality theory; Hahn–Wilson’s redshift theorem; and Pstrągowski’s large-prime algebraicity comparison. The distinct unrestricted telescope counterexamples appear in Burklund–Hahn–Levy–Schlank.

Source manuscripts

• M1. OpenAI, Counterexamples to the Hahn-Wilson conjecture at height two (September 26, 2026).

Sources consulted on 22 October 2026 at repository revision 1581d4c. Download the manuscript citations.