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The structure of the Bousfield lattice

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arxiv math/9801103 v1 pith:EZ6RY5OG submitted 1998-01-22 math.AT

Mark Hovey , John Palmieri This is my paper · ORCID
classification math.AT
keywords bousfieldalgebraconjectureslatticebooleanclassescompleteparticular
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Using Ohkawa's theorem that the collection of Bousfield classes is a set, we perform a number of constructions with Bousfield classes. In particular, we describe a greatest lower bound operator; we also note that a certain subset DL of the Bousfield lattice is a frame, and we examine some consequences of this observation. We make several conjectures about the structure of the Bousfield lattice and DL. In particular, we conjecture that DL is obtained by killing "strange" spectra, such as the Brown-Comenetz dual of the sphere. We introduce a new "Boolean algebra of spectra" cBA, which contains Bousfield's BA and is complete. Our conjectures allow us to identify cBA as being isomorphic to the complete atomic Boolean algebra on {K(n) : n>= 0}, {A(n) : n>= 2}, and HF_p. Our conjectures imply that BA is the subBoolean algebra consisting of finite wedges of the K(n) and A(n), and their complements.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Tensor Nilpotence and the Size of the Bousfield Lattice

    math.AT 2026-08 conditional novelty 8.0 of 10

    For each prime p and integer n, the author builds spectra X_n with X_n^{⊗n} ≠ 0 but X_n^{⊗(n+1)} ≃ 0, refuting the Hovey-Palmieri retract conjecture and showing the Bousfield lattice has 2^{2^{ℵ_0}} elements.

  2. A strong finiteness condition for smashing localisations

    math.AT 2025-09 conditional novelty 7.0 of 10

    The localisation L^f_n is compactly central, and the compactly central localisations of spectra are exactly the finite ones that are the identity away from finitely many primes.

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