The strongly dualizable generalized smashing conjecture in the BP-local category of spectra

Wolcott (2015) proved that in any well generated tensor-triangulated category whose tensor unit generates the whole category, localization away from a set of strongly dualizable objects is always a smashing localization, and posed the converse — the strongly dualizable generalized smashing conjecture (SDGSC) — for the localized categories of spectra he studies. The SDGSC is now known to fail in the unlocalized stable homotopy category, as a consequence of the 2023 disproof of Ravenel's telescope conjecture, and in derived categories of commutative rings; it has been verified in the harmonic, E(n)-local, K(n)-local, HFp-local, I-local, and $Sp_{k,n}$-local categories. The one case explicitly left open by the source paper survives untouched: this problem asks for a decision of the SDGSC in the BP-local category.

Background

Fix a prime $p$ and let $\mathcal{S}$ denote the $p$-local stable homotopy category of spectra, with smash product $\wedge$ and unit the $p$-local sphere spectrum $\mathbb{S}^0$. A Bousfield localization is an idempotent functor $L\colon\mathcal{S}\to\mathcal{S}$ together with a natural map $X\to LX$ initial among maps to $L$-local objects; it is smashing if it commutes with all coproducts, equivalently if $LX\simeq X\wedge L\mathbb{S}^0$ for every $X$. Chromatic homotopy theory compares two families of smashing localizations: the finite localization $L^f_n$, localization away from a finite type-$(n+1)$ spectrum $F(n+1)$ (a finite $p$-local spectrum of chromatic type $n+1$, i.e. its Morava $K(i)$-homology vanishes for $i\leq n$ and is nonzero for $i=n+1$), and the chromatic localization $L_n$ at the wedge $K(0)\vee\cdots\vee K(n)$ of Morava $K$-theories, the homology theories $K(i)_*(-)$ whose coefficient rings are graded fields, one for each chromatic height $i$. Ravenel's telescope conjecture $\mathrm{TC}_n$ asserts $L^f_n\cong L_n$ for all $n$; it was proved for $n\leq 1$ and disproved for all $n\geq 2$ and all primes by Burklund–Hahn–Levy–Schlank (2023), using $T(n+1)$-localized algebraic K-theory of ring spectra, where $T(n+1)$ is the telescope (mapping telescope of a periodic $v_{n+1}$-self-map) of a finite type-$(n+1)$ spectrum.

An object $X$ of a tensor-triangulated category is compact if every map from $X$ to a coproduct factors through a finite subcoproduct, and strongly dualizable if the natural map $DX\wedge Y\to F(X,Y)$, with $DX=F(X,\mathbb{1})$ the Spanier–Whitehead dual and $F(-,-)$ the function-object bifunctor, is an equivalence for all $Y$. Localization away from any set of compact objects is smashing; the generalized smashing conjecture (GSC), formulated in the early 1990s for derived categories $D(R)$ of commutative rings and since studied in general tensor-triangulated categories, asserts the converse: every smashing localization has kernel generated, as a localizing subcategory, by a set of compact objects. The GSC is false in $D(R)$ for a commutative-ring counterexample constructed in 1994 and for an infinite explicit family of valuation domains classified by Balchin–Tecklenburg (2025); in $D(R)$ the compact and the strongly dualizable objects coincide (both are the perfect complexes), so these examples equally refute the strongly dualizable variant. Wolcott (2015) proved that in a well generated tensor-triangulated category $\mathsf{T}$ (triangulated, with set-indexed coproducts and, in the sense of Neeman–Krause, a set of $\alpha$-small generators for some regular cardinal $\alpha$) with $\operatorname{loc}(\mathbb{1})=\mathsf{T}$ — the tensor unit generates $\mathsf{T}$ as a localizing subcategory — localization away from any set of strongly dualizable objects is smashing (his Theorem A), and posed the strongly dualizable generalized smashing conjecture (SDGSC): every smashing localization is localization away from a set of strongly dualizable objects. Wolcott explicitly restricts this conjecture to the topological categories he studies, noting that the 1994 ring counterexample already refutes it in complete generality.

For a spectrum $Z$, the $Z$-local category $\mathcal{L}_Z$, the full subcategory of $Z$-local spectra (those $X$ for which $X\to L_ZX$ is an equivalence), inherits a tensor-triangulated structure with unit $L_Z\mathbb{S}^0$; these categories are typically well generated but not compactly generated. Wolcott determined the landscape: the SDGSC holds in the harmonic category (local with respect to the wedge $\bigvee_{i\geq 0}K(i)$), in the $H\mathbb{F}_p$-local category ($H\mathbb{F}_p$ the Eilenberg–Mac Lane spectrum of the prime field), and in the $I$-local category ($I$ the Brown–Comenetz dual of the sphere, characterized by $\pi_{-*}(IX)\cong\operatorname{Hom}(\pi_*X,\mathbb{Q}/\mathbb{Z})$ for all spectra $X$), while the GSC fails in each of these because they contain no nonzero compact objects; both the GSC and the SDGSC hold in the $E(n)$- and $K(n)$-local categories, where $E(n)$ is Johnson–Wilson theory with $E(n)_*=\mathbb{Z}_{(p)}[v_1,\dots,v_n,v_n^{-1}]$. Heard (2023) extended the positive side to the interpolating $Sp_{k,n}$-local categories, the categories of spectra local with respect to $K(k)\vee\cdots\vee K(n)$, proving all localized telescope-conjecture variants together with the GSC and the SDGSC there.

The one case Wolcott leaves open is the BP-local category $\mathcal{L}_{BP}$, the category of spectra local with respect to the Brown–Peterson spectrum $BP$, the $p$-local ring spectrum with coefficient ring $BP_*=\mathbb{Z}_{(p)}[v_1,v_2,\dots]$ carrying the universal $p$-typical formal group law, which presents $p$-local complex cobordism and is central to chromatic homotopy theory. Wolcott proves the localized telescope conjecture there: for every $n$ the chromatic localizations $L_n$, $l_n$, and $L^f_n$ agree on $\mathcal{L}_{BP}$, giving a chain of distinct smashing localizations, each generated by a strongly dualizable object; but $\mathcal{L}_{BP}$ has no nonzero compact objects, so the GSC fails there (his Proposition 6.10), and whether every smashing localization on $\mathcal{L}_{BP}$ is generated by strongly dualizable objects is left open. His main tool for constraining smashing localizations is the Bousfield lattice: the set of Bousfield classes $\langle X\rangle=\{W\mid W\wedge X=0\}$, a set by Ohkawa-type results, ordered by inclusion; every smashing localization determines a complemented Bousfield class, and Wolcott bounds the Bousfield lattice of $\mathcal{L}_{BP}$ only by $2^{\aleph_0}\leq|\mathrm{BL}(\mathcal{L}_{BP})|\leq 2^{2^{\aleph_0}}$. He remarks that finding any smashing localization on $\mathcal{L}_{BP}$ beyond the chromatic ones is probably at least as hard as doing so on $\mathcal{S}$, where the classification of smashing localizations is itself open.

Two later developments frame the question. First, Wolcott's Theorem 3.2 shows that on $\mathcal{S}$ the GSC and SDGSC are equivalent and imply $\mathrm{TC2}_n$ for all $n$; since the telescope conjecture is now false at all heights $n\geq 2$, the GSC and SDGSC are refuted in $\mathcal{S}$ itself, so the surviving content of the SDGSC lies precisely in the non-compactly-generated localized categories. Second, the general frameworks for smashing subcategories — the frame of smashing tensor-ideals of Balmer–Krause–Stevenson (2020) and the smashing-spectrum characterization of the telescope conjecture of Verasdanis (2023) — are proved for rigidly-compactly generated categories and do not apply directly to $\mathcal{L}_{BP}$, which has no nonzero compact objects.

Problem Statement

Let $p$ be a prime and let $\mathcal{L}_{BP}$ denote the BP-local category of $p$-local spectra: the full subcategory of $BP$-local spectra with its induced tensor-triangulated structure. Decide the strongly dualizable generalized smashing conjecture (SDGSC) in $\mathcal{L}_{BP}$: is every smashing localization functor $l\colon\mathcal{L}_{BP}\to\mathcal{L}_{BP}$ localization away from a set of strongly dualizable objects, that is, does $\ker l=\operatorname{loc}(A)$ hold for some set $A$ of strongly dualizable objects of $\mathcal{L}_{BP}$? Here a localizing subcategory is a full triangulated subcategory closed under retracts and set-indexed coproducts, and $\operatorname{loc}(A)$ is the smallest localizing subcategory of $\mathcal{L}_{BP}$ containing $A$. The question is settled either by a proof that the SDGSC holds in $\mathcal{L}_{BP}$ for every prime $p$, or by an explicit smashing localization functor on $\mathcal{L}_{BP}$, for at least one prime $p$, whose kernel is not the localizing subcategory generated by any set of strongly dualizable objects of $\mathcal{L}_{BP}$.

References

1. F. Luke Wolcott, "Variations of the telescope conjecture and Bousfield lattices for localized categories of spectra", Pacific Journal of Mathematics 276 (2015), 483–509. DOI: 10.2140/pjm.2015.276.483; arXiv:1307.3351. https://arxiv.org/abs/1307.3351
2. Robert Burklund, Jeremy Hahn, Ishan Levy, Tomer M. Schlank, "$K$-theoretic counterexamples to Ravenel's telescope conjecture", arXiv:2310.17459 (2023). https://arxiv.org/abs/2310.17459
3. Drew Heard, "The $Sp_{k,n}$-local stable homotopy category", Algebraic & Geometric Topology 23 (2023), 3655–3706. DOI: 10.2140/agt.2023.23.3655; arXiv:2108.02486. https://arxiv.org/abs/2108.02486
4. Scott Balchin, Florian Tecklenburg, "Classifying smashing ideals in derived categories of valuation domains", Journal of Pure and Applied Algebra 229 (2025), article 107917. DOI: 10.1016/j.jpaa.2025.107917; arXiv:2407.11791. https://arxiv.org/abs/2407.11791
5. Paul Balmer, Henning Krause, Greg Stevenson, "The frame of smashing tensor-ideals", Mathematical Proceedings of the Cambridge Philosophical Society 168 (2020), 323–343. DOI: 10.1017/s0305004118000725; arXiv:1701.05937. https://arxiv.org/abs/1701.05937
6. Agnes Beaudry, Mark Behrens, Prasit Bhattacharya, Dominic Culver, Zhouli Xu, "The telescope conjecture at height 2 and the tmf resolution", Journal of Topology 14 (2021), 1243–1320. DOI: 10.1112/topo.12208; arXiv:1909.13379. https://arxiv.org/abs/1909.13379
7. Charalampos Verasdanis, "Stratification and the smashing spectrum", Mathematische Zeitschrift 305 (2023), article 54. DOI: 10.1007/s00209-023-03378-4; arXiv:2204.01082. https://arxiv.org/abs/2204.01082