Chromatic Picard groups at large primes

Pstrągowski, Piotr

doi:10.1090/proc/16004

Chromatic Picard groups at large primes
HTML articles powered by AMS MathViewer

by Piotr Pstrągowski PDF

Proc. Amer. Math. Soc. 150 (2022), 4981-4988 Request permission

Abstract:

We show that the Hopkins’ Picard group of the $K(n)$-local category coincides with its algebraic approximation when $2p-2 > n^{2}+n$.

References

Andrew Baker, $L$-complete Hopf algebroids and their comodules, Alpine perspectives on algebraic topology, Contemp. Math., vol. 504, Amer. Math. Soc., Providence, RI, 2009, pp. 1–22. MR 2581903, DOI 10.1090/conm/504/09873
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
Tobias Barthel and Drew Heard, The $E_2$-term of the $K(n)$-local $E_n$-Adams spectral sequence, Topology Appl. 206 (2016), 190–214. MR 3494442, DOI 10.1016/j.topol.2016.03.024
Tobias Barthel, Drew Heard, and Niko Naumann, On conjectures of Hovey-Strickland and Chai, Selecta Math. (N.S.) 28 (2022), no. 3, Paper No. 56, 31. MR 4405748, DOI 10.1007/s00029-022-00766-2
Andrew Baker and Birgit Richter, Invertible modules for commutative $\Bbb S$-algebras with residue fields, Manuscripta Math. 118 (2005), no. 1, 99–119. MR 2171294, DOI 10.1007/s00229-005-0582-1
Ethan S. Devinatz and Michael J. Hopkins, Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups, Topology 43 (2004), no. 1, 1–47. MR 2030586, DOI 10.1016/S0040-9383(03)00029-6
Paul Goerss and Michael Hopkins, Moduli spaces of commutative ring spectra, London Math. Soc. Lecture Note Ser. 315 (2005), 151–200.
P. Goerss, H.-W. Henn, M. Mahowald, and C. Rezk, A resolution of the $K(2)$-local sphere at the prime 3, Ann. of Math. (2) 162 (2005), no. 2, 777–822. MR 2183282, DOI 10.4007/annals.2005.162.777
Paul Goerss, Hans-Werner Henn, Mark Mahowald, and Charles Rezk, On Hopkins’ Picard groups for the prime 3 and chromatic level 2, J. Topol. 8 (2015), no. 1, 267–294. MR 3335255, DOI 10.1112/jtopol/jtu024
David Gepner and Tyler Lawson, Brauer groups and Galois cohomology of commutative ring spectra, Compos. Math. 157 (2021), no. 6, 1211–1264. MR 4270473, DOI 10.1112/s0010437x21007065
Drew Heard, Morava modules and the $K(n)$-local Picard group, Bull. Aust. Math. Soc. 92 (2015), no. 1, 171–172. MR 3366462, DOI 10.1017/S0004972715000507
Drew Heard,The $sp_{k, n}$-local stable homotopy category, Preprint, arXiv:2108.02486, 2021.
Michael J. Hopkins, Mark Mahowald, and Hal Sadofsky, Constructions of elements in Picard groups, Topology and representation theory (Evanston, IL, 1992) Contemp. Math., vol. 158, Amer. Math. Soc., Providence, RI, 1994, pp. 89–126. MR 1263713, DOI 10.1090/conm/158/01454
Mark Hovey, Some spectral sequences in Morava $E$-theory, Preprint 29 (2004).
Mark Hovey and Hal Sadofsky, Invertible spectra in the $E(n)$-local stable homotopy category, J. London Math. Soc. (2) 60 (1999), no. 1, 284–302. MR 1722151, DOI 10.1112/S0024610799007784
Mark Hovey and Neil P. Strickland, Morava $K$-theories and localisation, Mem. Amer. Math. Soc. 139 (1999), no. 666, viii+100. MR 1601906, DOI 10.1090/memo/0666
Jacob Lurie and Michael Hopkins, On Brauer groups of Lubin-Tate spectra I, March 30, 2017, http://www.math.harvard.edu/~lurie/papers/Brauer.pdf.
Akhil Mathew and Vesna Stojanoska, The Picard group of topological modular forms via descent theory, Geom. Topol. 20 (2016), no. 6, 3133–3217. MR 3590352, DOI 10.2140/gt.2016.20.3133
Piotr Pstrągowski, Chromatic homotopy theory is algebraic when $p>n^2+n+1$, Adv. Math. 391 (2021), Paper No. 107958, 37. MR 4300917, DOI 10.1016/j.aim.2021.107958
Piotr Pstrągowski and Paul VanKoughnett, Abstract Goerss-Hopkins theory, Adv. Math. 395 (2022), Paper No. 108098, 51. MR 4363589, DOI 10.1016/j.aim.2021.108098
John Rognes, Galois extensions of structured ring spectra. Stably dualizable groups, Mem. Amer. Math. Soc. 192 (2008), no. 898, viii+137. MR 2387923, DOI 10.1090/memo/0898
N. P. Strickland, On the $p$-adic interpolation of stable homotopy groups, Adams Memorial Symposium on Algebraic Topology, 2 (Manchester, 1990) London Math. Soc. Lecture Note Ser., vol. 176, Cambridge Univ. Press, Cambridge, 1992, pp. 45–54. MR 1232198, DOI 10.1017/CBO9780511526312.010