A relative Lubin–Tate theorem via higher formal geometry

Mazel-Gee, Aaron¹ ; Peterson, Eric¹ ; Stapleton, Nathaniel²

¹Department of Mathematics, University of California Berkeley, 970 Evans Hall #3840, Berkeley, CA 94720-3840, USA
²Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue E18-369, Cambridge, MA 02139-4307, USA

Algebraic and Geometric Topology, Tome 15 (2015) no. 4, pp. 2239-2268

Résumé

We formulate a theory of punctured affine formal schemes, suitable for describing certain phenomena within algebraic topology. As a proof-of-concept we show that the Morava K–theoretic localizations of Morava E–theory, which arise in transchromatic homotopy theory, corepresent a Lubin–Tate-type moduli problem in this framework.

Détail

DOI : 10.2140/agt.2015.15.2239

Classification : 14L05, 55N22
Keywords: formal group, deformation, transchromatic homotopy theory, Lubin–Tate space

Bibliographie

[1] M Ando, C P French, N Ganter, The Jacobi orientation and the two-variable elliptic genus, Algebr. Geom. Topol. 8 (2008) 493

[2] M Ando, J Morava, A renormalized Riemann–Roch formula and the Thom isomorphism for the free loop space, from: "Topology, geometry, and algebra: Interactions and new directions" (editors A Adem, G Carlsson, R Cohen), Contemp. Math. 279, Amer. Math. Soc. (2001) 11

[3] M Ando, J Morava, H Sadofsky, Completions of $\mathbb{Z}/(p)$–Tate cohomology of periodic spectra, Geom. Topol. 2 (1998) 145

[4] M Artin, Versal deformations and algebraic stacks, Invent. Math. 27 (1974) 165

[5] A A Beĭlinson, Residues and adèles, Funktsional. Anal. i Prilozhen. 14 (1980) 44

[6] J P C Greenlees, N P Strickland, Varieties and local cohomology for chromatic group cohomology rings, Topology 38 (1999) 1093

[7] M Hovey, Bousfield localization functors and Hopkins' chromatic splitting conjecture, from: "The Čech centennial" (editors M Cenkl, H Miller), Contemp. Math. 181, Amer. Math. Soc. (1995) 225

[8] A Huber, On the Parshin–Beĭlinson adèles for schemes, Abh. Math. Sem. Univ. Hamburg 61 (1991) 249

[9] D C Isaksen, A model structure on the category of prosimplicial sets, Trans. Amer. Math. Soc. 353 (2001) 2805

[10] D C Isaksen, Calculating limits and colimits in procategories, Fund. Math. 175 (2002) 175

[11] P T Johnstone, Stone spaces, Cambridge Studies in Advanced Math. 3, Cambridge Univ. Press (1982)

[12] K Kato, Existence theorem for higher local fields, from: "Invitation to higher local fields" (editors I Fesenko, M Kurihara), Geom. Topol. Monogr. 3 (2000) 165

[13] K Kato, S Saito, Global class field theory of arithmetic schemes, from: "Applications of algebraic $K$–theory to algebraic geometry and number theory, Part I, II" (editors S J Bloch, R K Dennis, E M Friedlander, M R Stein), Contemp. Math. 55, Amer. Math. Soc. (1986) 255

[14] T Lawson, N Naumann, Commutativity conditions for truncated Brown–Peterson spectra of height $2$, J. Topol. 5 (2012) 137

[15] J Lubin, J Tate, Formal moduli for one-parameter formal Lie groups, Bull. Soc. Math. France 94 (1966) 49

[16] D V Osipov, The Krichever correspondence for algebraic varieties, Izv. Ross. Akad. Nauk Ser. Mat. 65 (2001) 91

[17] D V Osipov, $n$–dimensional local fields and adèles on $n$–dimensional schemes, from: "Surveys in contemporary mathematics" (editors N Young, Y Choi), London Math. Soc. Lecture Note Ser. 347, Cambridge Univ. Press (2008) 131

[18] C Rezk, Notes on the Hopkins–Miller theorem, from: "Homotopy theory via algebraic geometry and group representations" (editors M Mahowald, S Priddy), Contemp. Math. 220, Amer. Math. Soc. (1998) 313

[19] S Saito, Arithmetic on two-dimensional local rings, Invent. Math. 85 (1986) 379

[20] S Saito, Class field theory for two-dimensional local rings, from: "Galois representations and arithmetic algebraic geometry" (editor Y Ihara), Adv. Stud. Pure Math. 12, North-Holland (1987) 343

[21] T M Schlank, N Stapleton, A transchromatic proof of Strickland's theorem,

[22] N Stapleton, Transchromatic generalized character maps, Algebr. Geom. Topol. 13 (2013) 171

[23] N Stapleton, Transchromatic twisted character maps, J. Homotopy Relat. Struct. 10 (2015) 29

[24] T Torii, On degeneration of one-dimensional formal group laws and applications to stable homotopy theory, Amer. J. Math. 125 (2003) 1037

[25] T Torii, Milnor operations and the generalized Chern character, from: "Proceedings of the Nishida Fest" (editors M Ando, N Minami, J Morava, W S Wilson), Geom. Topol. Monogr. 10 (2007) 383

[26] T Torii, Comparison of Morava $E$–theories, Math. Zeitschrift 266 (2010) 933

[27] T Torii, HKR characters, $p$–divisible groups and the generalized Chern character, Trans. Amer. Math. Soc. 362 (2010) 6159

[28] T Torii, $K(n)$–localization of the $K(n+1)$–local $E_{n+1}$–Adams spectral sequences, Pacific J. Math. 250 (2011) 439