Coalgebraic formal curve spectra and spectral jet spaces

Peterson, Eric¹

¹ Department of Mathematics, Harvard University, Cambridge, MA, United States

Geometry & topology, Tome 24 (2020) no. 1, pp. 1-47.

Résumé

We import into homotopy theory the algebrogeometric construction of the cotangent space of a geometric point on a scheme. Specializing to the category of spectra local to a Morava $K$–theory of height $d$, we show that this can be used to produce a choice-free model of the determinantal sphere as well as an efficient Picard-graded cellular decomposition of $K\left({\mathbb{Z}}_p,d+1\right)$. Coupling these ideas to work of Westerland, we give a “Snaith’s theorem” for the Iwasawa extension of the $K\left(d\right)$–local sphere.

Détail

DOI : 10.2140/gt.2020.24.1

Classification : 55N22, 55P20, 55P60
Keywords: chromatic homotopy, formal group, Morava $E$–theory, determinantal sphere, inverse limit, comodule

Bibliographie

[1] M F Atiyah, Thom complexes, Proc. London Math. Soc. 11 (1961) 291 | DOI

[2] T Barthel, A Beaudry, P G Goerss, V Stojanoska, Constructing the determinant sphere using a Tate twist, preprint (2018)

[3] T Barthel, A Beaudry, E Peterson, H Sadofsky, The homology of inverse limits and the algebraic chromatic splitting conjecture, unpublished (2017)

[4] M Behrens, A modular description of the K(2)–local sphere at the prime 3, Topology 45 (2006) 343 | DOI

[5] M Behrens, The homotopy groups of SE(2) at p ≥ 5 revisited, Adv. Math. 230 (2012) 458 | DOI

[6] J M Boardman, Conditionally convergent spectral sequences, from: "Homotopy invariant algebraic structures" (editors J P Meyer, J Morava, W S Wilson), Contemp. Math. 239, Amer. Math. Soc. (1999) 49 | DOI

[7] A K Bousfield, The localization of spectra with respect to homology, Topology 18 (1979) 257 | DOI

[8] V Buchstaber, A Lazarev, Dieudonné modules and p–divisible groups associated with Morava K–theory of Eilenberg–Mac Lane spaces, Algebr. Geom. Topol. 7 (2007) 529 | DOI

[9] M Ching, Bar constructions for topological operads and the Goodwillie derivatives of the identity, Geom. Topol. 9 (2005) 833 | DOI

[10] M Demazure, Lectures on p–divisible groups, 302, Springer (1986)

[11] E S Devinatz, Towards the finiteness of π∗LK(n)S0, Adv. Math. 219 (2008) 1656 | DOI

[12] E S Devinatz, M J Hopkins, Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups, Topology 43 (2004) 1 | DOI

[13] V Ginzburg, M Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994) 203 | DOI

[14] P G Goerss, Hopf rings, Dieudonné modules, and E∗Ω2S3, from: "Homotopy invariant algebraic structures" (editors J P Meyer, J Morava, W S Wilson), Contemp. Math. 239, Amer. Math. Soc. (1999) 115 | DOI

[15] P Goerss, H W Henn, M Mahowald, C Rezk, On Hopkins’ Picard groups for the prime 3 and chromatic level 2, J. Topol. 8 (2015) 267 | DOI

[16] R Haugseng, The higher Morita category of En–algebras, Geom. Topol. 21 (2017) 1631 | DOI

[17] S M H Hedayatzadeh, Exterior powers of π–divisible modules over fields, J. Number Theory 138 (2014) 119 | DOI

[18] S M H Hedayatzadeh, Exterior powers of Lubin–Tate groups, J. Théor. Nombres Bordeaux 27 (2015) 77

[19] M J Hopkins, B H Gross, The rigid analytic period mapping, Lubin–Tate space, and stable homotopy theory, Bull. Amer. Math. Soc. 30 (1994) 76 | DOI

[20] M J Hopkins, J Lurie, Ambidexterity in K(n)–local stable homotopy theory, preprint (2013)

[21] M J Hopkins, M Mahowald, H Sadofsky, Constructions of elements in Picard groups, from: "Topology and representation theory" (editors E M Friedlander, M E Mahowald), Contemp. Math. 158, Amer. Math. Soc. (1994) 89 | DOI

[22] M J Hopkins, J H Smith, Nilpotence and stable homotopy theory, II, Ann. of Math. 148 (1998) 1 | DOI

[23] M Hovey, The generalized homology of products, Glasg. Math. J. 49 (2007) 1 | DOI

[24] M Hovey, N P Strickland, Morava K–theories and localisation, 666, Amer. Math. Soc. (1999) | DOI

[25] D C Johnson, W S Wilson, The Brown–Peterson homology of elementary p–groups, Amer. J. Math. 107 (1985) 427 | DOI

[26] J Lurie, Higher algebra, book project (2017)

[27] E Peterson, Report on E–theory conjectures seminar, seminar notes (2013)

[28] D C Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1984) 351 | DOI

[29] D C Ravenel, Complex cobordism and stable homotopy groups of spheres, 121, Academic (1986)

[30] D C Ravenel, Nilpotence and periodicity in stable homotopy theory, 128, Princeton Univ. Press (1992)

[31] D C Ravenel, W S Wilson, The Morava K–theories of Eilenberg–Mac Lane spaces and the Conner–Floyd conjecture, Amer. J. Math. 102 (1980) 691 | DOI

[32] H Sadofsky, The homology of inverse limits of spectra, preprint (2001)

[33] H Sadofsky, Towards the chromatic splitting conjecture, preprint (2003)

[34] V Snaith, Localized stable homotopy of some classifying spaces, Math. Proc. Cambridge Philos. Soc. 89 (1981) 325 | DOI

[35] V Stojanoska, Duality for topological modular forms, Doc. Math. 17 (2012) 271

[36] N P Strickland, A conjecture about the K(n)–local Picard group, unpublished

[37] N P Strickland, On the p–adic interpolation of stable homotopy groups, from: "Adams Memorial Symposium on Algebraic Topology, II" (editors N Ray, G Walker), London Math. Soc. Lecture Note Ser. 176, Cambridge Univ. Press (1992) 45 | DOI

[38] N P Strickland, Formal schemes and formal groups, from: "Homotopy invariant algebraic structures" (editors J P Meyer, J Morava, W S Wilson), Contemp. Math. 239, Amer. Math. Soc. (1999) 263 | DOI

[39] N P Strickland, Gross–Hopkins duality, Topology 39 (2000) 1021 | DOI

[40] D P Sullivan, Geometric topology : localization, periodicity and Galois symmetry, 8, Springer (2005)

[41] C Westerland, A higher chromatic analogue of the image of J, Geom. Topol. 21 (2017) 1033 | DOI