@incollection{hovey1995,
  author = {Hovey, Mark},
  title = {Bousfield localization functors and {Hopkins}' chromatic splitting conjecture},
  booktitle = {The {Cech} Centennial},
  series = {Contemporary Mathematics},
  volume = {181},
  pages = {225--250},
  publisher = {American Mathematical Society},
  year = {1995},
  doi = {10.1090/conm/181/02036},
  note = {Conjecture numbering follows the July 1993 author manuscript. \href{https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/chromatic-splitting.pdf}{Cited text}. \href{https://doi.org/10.1090/conm/181/02036}{doi:10.1090/conm/181/02036}},
  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/chromatic-splitting.pdf},
}

@article{beaudry2017,
  author = {Beaudry, Agn{\`e}s},
  title = {The chromatic splitting conjecture at {$n=p=2$}},
  journal = {Geometry \& Topology},
  volume = {21},
  pages = {3213--3230},
  year = {2017},
  doi = {10.2140/gt.2017.21.3213},
  note = {Theorem 1.0.4 is numbered in \href{https://arxiv.org/abs/1502.02190v3}{arXiv:1502.02190v3}; the published theorem is numbered 1.4. \href{https://doi.org/10.2140/gt.2017.21.3213}{doi:10.2140/gt.2017.21.3213}},
}

@article{bgh2022,
  author = {Beaudry, Agn{\`e}s and Goerss, Paul G. and Henn, Hans-Werner},
  title = {Chromatic splitting for the {$K(2)$}-local sphere at {$p=2$}},
  journal = {Geometry \& Topology},
  volume = {26},
  pages = {377--476},
  year = {2022},
  doi = {10.2140/gt.2022.26.377},
  note = {Statement numbering follows \href{https://arxiv.org/abs/1712.08182v4}{arXiv:1712.08182v4}, revised 6 October 2020. \href{https://doi.org/10.2140/gt.2022.26.377}{doi:10.2140/gt.2022.26.377}},
}

@article{dh2004,
  author = {Devinatz, Ethan S. and Hopkins, Michael J.},
  title = {Homotopy fixed point spectra for closed subgroups of the {Morava} stabilizer groups},
  journal = {Topology},
  volume = {43},
  number = {1},
  pages = {1--47},
  year = {2004},
  doi = {10.1016/S0040-9383(03)00029-6},
  note = {\href{https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/dh04.pdf}{Cited text}. \href{https://doi.org/10.1016/S0040-9383(03)00029-6}{doi:10.1016/S0040-9383(03)00029-6}},
}

@article{hs1999,
  author = {Hovey, Mark and Strickland, Neil P.},
  title = {{Morava} {$K$}-theories and localisation},
  journal = {Memoirs of the American Mathematical Society},
  volume = {139},
  number = {666},
  year = {1999},
  doi = {10.1090/memo/0666},
  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/kn.pdf},
  note = {Statement locators use the author-format text. \href{https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/kn.pdf}{Cited text}. \href{https://doi.org/10.1090/memo/0666}{doi:10.1090/memo/0666}},
}

@article{hs-comodules2003,
  author = {Hovey, Mark and Strickland, Neil P.},
  title = {Comodules and {Landweber} exact homology theories},
  journal = {Advances in Mathematics},
  volume = {192},
  number = {2},
  pages = {427--456},
  year = {2005},
  doi = {10.1016/j.aim.2004.04.011},
  note = {Statement numbering follows \href{https://arxiv.org/abs/math/0301232v1}{arXiv:math/0301232v1}. \href{https://doi.org/10.1016/j.aim.2004.04.011}{doi:10.1016/j.aim.2004.04.011}},
}

@article{rezk2009,
  author = {Rezk, Charles},
  title = {The congruence criterion for power operations in {Morava} {$E$}-theory},
  journal = {Homology, Homotopy and Applications},
  volume = {11},
  number = {2},
  pages = {327--379},
  year = {2009},
  note = {Cited version: \href{https://arxiv.org/abs/0902.2499v2}{arXiv:0902.2499v2}},
}

@article{bbs2020,
  author = {Barthel, Tobias and Beaudry, Agn{\`e}s and Stojanoska, Vesna},
  title = {{Gross--Hopkins} duals of higher real {$K$}-theory spectra},
  journal = {Transactions of the American Mathematical Society},
  volume = {372},
  number = {5},
  pages = {3347--3368},
  year = {2019},
  note = {Cited version:
 \href{https://arxiv.org/abs/1705.07036v2}{arXiv:1705.07036v2},
 revised 16 June 2020. \href{https://doi.org/10.1090/tran/7730}{doi:10.1090/tran/7730}},
  doi = {10.1090/tran/7730},
}

@misc{heard2015,
  author = {Heard, Drew},
  title = {The {Tate} spectrum of the higher real {$K$}-theories at height {$n=p-1$}},
  year = {2015},
  note = {Cited version: \href{https://arxiv.org/abs/1501.07759v2}{arXiv:1501.07759v2}, submitted 18 March 2015; the PDF is dated 19 March 2015},
}

@article{hms2017,
  author = {Heard, Drew and Mathew, Akhil and Stojanoska, Vesna},
  title = {Picard groups of higher real {$K$}-theory spectra at height {$p-1$}},
  journal = {Compositio Mathematica},
  volume = {153},
  number = {9},
  pages = {1820--1854},
  year = {2017},
  doi = {10.1112/S0010437X17007242},
  note = {Cited version: \href{https://arxiv.org/abs/1511.08064v1}{arXiv:1511.08064v1}. \href{https://doi.org/10.1112/S0010437X17007242}{doi:10.1112/S0010437X17007242}},
}

@article{mathew2016,
  author = {Mathew, Akhil},
  title = {The {Galois} group of a stable homotopy theory},
  journal = {Advances in Mathematics},
  volume = {291},
  pages = {403--541},
  year = {2016},
  doi = {10.1016/j.aim.2015.12.017},
  note = {\href{https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/Mathew-Galois.pdf}{Cited text}. \href{https://doi.org/10.1016/j.aim.2015.12.017}{doi:10.1016/j.aim.2015.12.017}},
}

@article{mnn2017,
  author = {Mathew, Akhil and Naumann, Niko and Noel, Justin},
  title = {Nilpotence and descent in equivariant stable homotopy theory},
  journal = {Advances in Mathematics},
  volume = {305},
  pages = {994--1084},
  year = {2017},
  doi = {10.1016/j.aim.2016.09.027},
  note = {\href{https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/MMN17.pdf}{Cited text}. \href{https://doi.org/10.1016/j.aim.2016.09.027}{doi:10.1016/j.aim.2016.09.027}},
}

@article{mnn2019,
  author = {Mathew, Akhil and Naumann, Niko and Noel, Justin},
  title = {Derived induction and restriction theory},
  journal = {Geometry \& Topology},
  volume = {23},
  number = {2},
  pages = {541--636},
  year = {2019},
  doi = {10.2140/gt.2019.23.541},
  note = {Citations use the published numbering. \href{https://msp.org/gt/2019/23-2/gt-v23-n2-p01-p.pdf}{Cited text}. \href{https://doi.org/10.2140/gt.2019.23.541}{doi:10.2140/gt.2019.23.541}},
}

@article{serre1965,
  author = {Serre, Jean-Pierre},
  title = {Sur la dimension cohomologique des groupes profinis},
  journal = {Topology},
  volume = {3},
  pages = {413--420},
  year = {1965},
  doi = {10.1016/0040-9383(65)90006-6},
  note = {\href{https://doi.org/10.1016/0040-9383(65)90006-6}{doi:10.1016/0040-9383(65)90006-6}},
}

@article{morava1985,
  author = {Morava, Jack},
  title = {Noetherian localisations of categories of cobordism comodules},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {121},
  pages = {1--39},
  year = {1985},
  doi = {10.2307/1971192},
  note = {\href{https://doi.org/10.2307/1971192}{doi:10.2307/1971192}},
}

@article{basterra-mandell2013,
  author = {Basterra, Maria and Mandell, Michael A.},
  title = {The multiplication on {$BP$}},
  journal = {Journal of Topology},
  volume = {6},
  pages = {285--310},
  year = {2013},
  doi = {10.1112/jtopol/jts032},
  url = {https://arxiv.org/abs/1101.0023v1},
  note = {Theorem locators use arXiv:1101.0023v1. \href{https://arxiv.org/abs/1101.0023v1}{Cited text}. \href{https://doi.org/10.1112/jtopol/jts032}{doi:10.1112/jtopol/jts032}},
}

@article{gross-hopkins1994,
  author = {Hopkins, Michael J. and Gross, Benedict H.},
  title = {The rigid analytic period mapping, {Lubin--Tate} space, and stable homotopy theory},
  journal = {Bulletin of the American Mathematical Society},
  volume = {30},
  pages = {76--86},
  year = {1994},
  doi = {10.1090/S0273-0979-1994-00438-0},
  note = {Theorem numbering follows \href{https://arxiv.org/abs/math/9401220v1}{arXiv:math/9401220v1}. \href{https://doi.org/10.1090/S0273-0979-1994-00438-0}{doi:10.1090/S0273-0979-1994-00438-0}},
}

@article{serre1951,
  author = {Serre, Jean-Pierre},
  title = {Homologie singuli{\`e}re des espaces fibr{\'e}s. Applications},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {54},
  number = {3},
  pages = {425--505},
  year = {1951},
  doi = {10.2307/1969485},
  note = {\href{https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/Serre-HSEF.pdf}{Cited journal scan}. \href{https://doi.org/10.2307/1969485}{doi:10.2307/1969485}},
}

@article{lubin-tate1966,
  author = {Lubin, Jonathan and Tate, John},
  title = {Formal moduli for one-parameter formal {Lie} groups},
  journal = {Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France},
  volume = {94},
  pages = {49--59},
  year = {1966},
  doi = {10.24033/bsmf.1633},
  note = {\href{https://doi.org/10.24033/bsmf.1633}{doi:10.24033/bsmf.1633}},
}

@incollection{rezk1998,
  author = {Rezk, Charles},
  title = {Notes on the {Hopkins--Miller} theorem},
  booktitle = {Homotopy Theory via Algebraic Geometry and Group Representations},
  series = {Contemporary Mathematics},
  volume = {220},
  pages = {313--366},
  publisher = {American Mathematical Society},
  year = {1998},
  doi = {10.1090/conm/220/03107},
  url = {https://rezk.web.illinois.edu/hopkins-miller-thm.pdf},
  note = {Statement locators use the author text, especially Section 15. \href{https://rezk.web.illinois.edu/hopkins-miller-thm.pdf}{Cited text}. \href{https://doi.org/10.1090/conm/220/03107}{doi:10.1090/conm/220/03107}},
}

@article{hovey2004,
  author = {Hovey, Mark},
  title = {Operations and co-operations in {Morava} {$E$}-theory},
  journal = {Homology, Homotopy and Applications},
  volume = {6},
  number = {1},
  pages = {201--236},
  year = {2004},
  doi = {10.4310/HHA.2004.v6.n1.a13},
  url = {https://emis.de/ft/51790},
  note = {Citations use the journal text available through EMIS. \href{https://emis.de/ft/51790}{Cited text}. \href{https://doi.org/10.4310/HHA.2004.v6.n1.a13}{doi:10.4310/HHA.2004.v6.n1.a13}},
}

@article{christensen-strickland1998,
  author = {Christensen, J. Daniel and Strickland, Neil P.},
  title = {Phantom maps and homology theories},
  journal = {Topology},
  volume = {37},
  number = {2},
  pages = {339--364},
  year = {1998},
  doi = {10.1016/S0040-9383(97)00031-1},
  note = {Statement locators use the
 \href{https://jdc.math.uwo.ca/papers/phantoms.pdf}
 {author version dated 16 November 2000}. \href{https://doi.org/10.1016/S0040-9383(97)00031-1}{doi:10.1016/S0040-9383(97)00031-1}},
}

@article{ravenel1978,
  author = {Ravenel, Douglas C.},
  title = {The non-existence of odd primary {Arf} invariant elements in stable homotopy},
  journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
  volume = {83},
  number = {3},
  pages = {429--443},
  year = {1978},
  doi = {10.1017/S0305004100054712},
  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/mypapers/arf_odd.pdf},
  note = {\href{https://doi.org/10.1017/S0305004100054712}{doi:10.1017/S0305004100054712}},
}

@article{lee-ravenel1994,
  author = {Lee, Chun-Nip and Ravenel, Douglas C.},
  title = {On the nilpotence order of {$\beta_1$}},
  journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
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  number = {3},
  pages = {483--488},
  year = {1994},
  doi = {10.1017/S0305004100072248},
  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/mypapers/lee.pdf},
  note = {Cochain locators follow the author preprint dated April 1993. \href{https://www.sas.rochester.edu/mth/sites/doug-ravenel/mypapers/lee.pdf}{Cited text}. \href{https://doi.org/10.1017/S0305004100072248}{doi:10.1017/S0305004100072248}},
}

@incollection{gh2004,
  author = {Goerss, Paul G. and Hopkins, Michael J.},
  title = {Moduli spaces of commutative ring spectra},
  booktitle = {Structured Ring Spectra},
  editor = {Baker, Andrew and Richter, Birgit},
  series = {London Mathematical Society Lecture Note Series},
  volume = {315},
  pages = {151--200},
  publisher = {Cambridge University Press},
  year = {2004},
  doi = {10.1017/CBO9780511529955.009},
  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/sum.pdf},
  note = {Statement locators use the author preprint; Corollaries 7.6--7.7 are on printed manuscript pp. 48--49. \href{https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/sum.pdf}{Cited text}. \href{https://doi.org/10.1017/CBO9780511529955.009}{doi:10.1017/CBO9780511529955.009}},
}

@article{minami2003,
  author = {Minami, Norihiko},
  title = {On the chromatic tower},
  journal = {American Journal of Mathematics},
  volume = {125},
  number = {3},
  pages = {449--473},
  year = {2003},
  doi = {10.1353/ajm.2003.0018},
  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/minami4.pdf},
  note = {\href{https://doi.org/10.1353/ajm.2003.0018}{doi:10.1353/ajm.2003.0018}},
}

@article{landweber1976,
  author = {Landweber, Peter S.},
  title = {Homological properties of comodules over {$MU_*(MU)$} and {$BP_*(BP)$}},
  journal = {American Journal of Mathematics},
  volume = {98},
  number = {3},
  pages = {591--610},
  year = {1976},
  doi = {10.2307/2373808},
  note = {\href{https://doi.org/10.2307/2373808}{doi:10.2307/2373808}},
}

@article{bobkova2025,
  author = {Bobkova, Irina and Lachmann, Andrea and Li, Ang and Lima, Alicia and Stojanoska, Vesna and Zhang, Adela YiYu},
  title = {Bounding the {$K(p-1)$}-local exotic {Picard} group at {$p>3$}},
  journal = {Topology and its Applications},
  volume = {376},
  pages = {109445},
  year = {2025},
  doi = {10.1016/j.topol.2025.109445},
  url = {https://arxiv.org/abs/2403.15572v2},
  note = {Cited version: \href{https://arxiv.org/abs/2403.15572v2}{arXiv:2403.15572v2}, submitted 2 July 2024. \href{https://doi.org/10.1016/j.topol.2025.109445}{doi:10.1016/j.topol.2025.109445}},
}

@misc{goerss-hopkins2020,
  author = {Goerss, Paul G. and Hopkins, Michael J.},
  title = {Comparing dualities in the {$K(n)$}-local category},
  year = {2020},
  note = {Cited version: \href{https://arxiv.org/abs/2011.02011v1}{arXiv:2011.02011v1}, submitted 3 November 2020},
  url = {https://arxiv.org/abs/2011.02011v1},
}

@misc{stacks,
  author = {{The Stacks Project Authors}},
  title = {The {Stacks Project}},
  year = {2026},
  url = {https://stacks.math.columbia.edu},
  note = {\href{https://stacks.math.columbia.edu/tag/00O7}{Tag 00O7}
    and \href{https://stacks.math.columbia.edu/tag/00IN}{Tag 00IN}}
}