Redshift and multiplication for truncated Brown–Peterson spectra

Abstract

We equip $\mathrm {BP} \langle n \rangle $ with an $\mathbb {E}_3$-$\mathrm{BP}$-algebra structure for each prime $p$ and height $n$. The algebraic $K$-theory of this ring is of chromatic height exactly $n+1$, and the map $\mathrm {K}(\mathrm{BP}\langle n \rangle )_{(p)} \to \mathrm {L}_{n+1}^{f} \mathrm {K}(\mathrm {BP}\langle n\rangle )_{(p)}$ has bounded above fiber.

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      author = {Lunøe-Nielsen, Sverre and Rognes, John},
      title = {The {S}egal conjecture for topological {H}ochschild homology of complex cobordism},
      journal = {J. Topol.},
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      volume = {4},
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      pages = {591--622},
      issn = {1753-8416},
      mrclass = {55P91 (55N22 55P43 55S10)},
      mrnumber = {2832570},
      mrreviewer = {Birgit Richter},
      doi = {10.1112/jtopol/jtr015},
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      author = {Lurie, J.},
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      author = {Lurie, J.},
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      author = {Mahowald, Mark and Rezk, Charles},
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      volume = {121},
      year = {1999},
      number = {6},
      pages = {1153--1177},
      issn = {0002-9327},
      mrclass = {55P42 (55N22 55P60 55S10 55T15)},
      mrnumber = {1719751},
      mrreviewer = {J. P. C. Greenlees},
      doi = {10.1353/ajm.1999.0043},
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      author = {Mathew, Akhil},
      title = {Examples of descent up to nilpotence},
      booktitle = {Geometric and {T}opological {A}spects of the {R}epresentation {T}heory of {F}inite {G}roups},
      series = {Springer Proc. Math. Stat.},
      volume = {242},
      pages = {269--311},
      publisher = {Springer, Cham},
      year = {2018},
      mrclass = {18G80},
      mrnumber = {3901164},
      mrreviewer = {Paul Balmer},
      doi = {10.1007/978-3-319-94033-5_11},
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      issn = {0010-437X},
      mrclass = {19D55 (55P42)},
      mrnumber = {4256236},
      doi = {10.1112/S0010437X21007144},
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      journal = {J. Topol.},
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      mrclass = {55P43 (55N22 55T15)},
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      title = {Problems in infinite loop space theory},
      booktitle = {Conference on {H}omotopy {T}heory ({E}vanston, {I}ll., 1974)},
      series = {Notas Mat. Simpos.},
      volume = {1},
      pages = {111--125},
      publisher = {Soc. Mat. Mexicana, México},
      year = {1975},
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      mrreviewer = {Harvey Margolis},
      doi = {10.1016/0022-4049(81)90104-3},
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      volume = {3},
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      mrclass = {55N20 (19M05 55Q50)},
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      author = {Quillen, Daniel},
      title = {Higher algebraic {$\mathrm{K}$}-theory},
      booktitle = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. 1},
      venue = {{V}ancouver, {B}. {C}., 1974},
      pages = {171--176},
      year = {1975},
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      mrnumber = {0422392},
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      author = {Ravenel, Douglas C.},
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      series = {Pure Appl. Math.},
      volume = {121},
      publisher = {Academic Press, Inc., Orlando, FL},
      year = {1986},
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      mrclass = {55-02 (55Qxx 57-02)},
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      author = {Ravenel, Douglas C.},
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      series = {Annals of Mathematics Studies},
      volume = {128},
      note = {Appendix C by Jeff Smith},
      publisher = {Princeton Univ. Press, Princeton, NJ},
      year = {1992},
      pages = {xiv+209},
      isbn = {0-691-02572-X},
      mrclass = {55P42 (55N22 55Q10 57R77)},
      mrnumber = {1192553},
      mrreviewer = {N. J. Kuhn},
      doi = {10.1515/9781400882489},
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      volume = {6},
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      mrnumber = {2199461},
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      doi = {10.2140/agt.2006.6.287},
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      title = {The {L}ichtenbaum-{Q}uillen conjecture for {$K/\ell_\ast[\beta^{-1}]$}},
      booktitle = {Current {T}rends in {A}lgebraic {T}opology, {P}art 1 ({L}ondon, {O}nt., 1981)},
      series = {CMS Conf. Proc.},
      volume = {2},
      pages = {117--139},
      publisher = {Amer. Math. Soc., Providence, R.I.},
      year = {1982},
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      }

Authors

Jeremy Hahn

Massachusetts Institute of Technology, Cambridge, MA

Dylan Wilson

Harvard University, Cambridge, MA

Current address:

West Virginia University, Morgantown, WV