Algebraic K-theory of elliptic cohomology

Angelini-Knoll, Gabriel¹ ; Ausoni, Christian¹ ; Culver, Dominic Leon² ; Höning, Eva³ ; Rognes, John⁴

¹Université Sorbonne Paris Nord, LAGA, CNRS, UMR 7539, Villetaneuse, France
²Max Planck Institute for Mathematics, Bonn, Germany
³Department of Mathematics, Radboud University, Nijmegen, The Netherlands
⁴Department of Mathematics, University of Oslo, Oslo, Norway

Geometry & topology, Tome 29 (2025) no. 2, pp. 619-686

Résumé

We calculate the mod-(p,v1,v2) homotopy V (2)∗TC ⁡ (BP ⁡ 〈2〉) of the topological cyclic homology of the truncated Brown–Peterson spectrum BP ⁡ 〈2〉, at all primes p ≥ 7, and show that it is a finitely generated and free 𝔽p[v3]-module on 12p + 4 generators in explicit degrees within the range − 1 ≤∗≤ 2p3 + 2p2 + 2p − 3. At these primes BP ⁡ 〈2〉 is a form of elliptic cohomology, and our result also determines the mod-(p,v1,v2) homotopy of its algebraic K-theory. Our computation is the first that exhibits chromatic redshift from pure v2-periodicity to pure v3-periodicity in a precise quantitative manner.

Détail

DOI : 10.2140/gt.2025.29.619

Keywords: algebraic $K$-theory, cyclotomic trace map, topological cyclic homology, elliptic cohomology, truncated Brown–Peterson spectrum, Smith–Toda complex, $v_n$-periodicity, chromatic redshift, power operation, homotopy fixed point, Tate construction

Bibliographie

[1] G Angelini-Knoll, D L Culver, E Höning, Topological Hochschild homology of truncated Brown–Peterson spectra, I, Algebr. Geom. Topol. 24 (2024) 2509 | DOI

[2] V Angeltveit, J Rognes, Hopf algebra structure on topological Hochschild homology, Algebr. Geom. Topol. 5 (2005) 1223 | DOI

[3] S Araki, H Toda, Multiplicative structures in modq cohomology theories, I, Osaka Math. J. 2 (1965) 71

[4] C Ausoni, Topological Hochschild homology of connective complex K-theory, Amer. J. Math. 127 (2005) 1261 | DOI

[5] C Ausoni, On the algebraic K-theory of the complex K-theory spectrum, Invent. Math. 180 (2010) 611 | DOI

[6] C Ausoni, J Rognes, Algebraic K-theory of topological K-theory, Acta Math. 188 (2002) 1 | DOI

[7] C Ausoni, J Rognes, The chromatic red-shift in algebraic K-theory, Enseign. Math. 54 (2008) 13 | DOI

[8] C Ausoni, J Rognes, Algebraic K-theory of the first Morava K-theory, J. Eur. Math. Soc. 14 (2012) 1041 | DOI

[9] A Baker, A Jeanneret, Brave new Hopf algebroids and extensions of MU-algebras, Homology Homotopy Appl. 4 (2002) 163 | DOI

[10] M G Barratt, M E Mahowald, M C Tangora, Some differentials in the Adams spectral sequence, II, Topology 9 (1970) 309 | DOI

[11] M Basterra, M A Mandell, The multiplication on BP, J. Topol. 6 (2013) 285 | DOI

[12] B Bhatt, M Morrow, P Scholze, Topological Hochschild homology and integral p-adic Hodge theory, Publ. Math. Inst. Hautes Études Sci. 129 (2019) 199 | DOI

[13] M Bökstedt, I Madsen, Topological cyclic homology of the integers, from: "-theory", Astérisque 226, Soc. Math. France (1994) 57

[14] M Bökstedt, I Madsen, Algebraic K-theory of local number fields : the unramified case, from: "Prospects in topology" (editor F Quinn), Ann. of Math. Stud. 138, Princeton Univ. Press (1995) 28 | DOI

[15] M Bökstedt, W C Hsiang, I Madsen, The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 111 (1993) 465 | DOI

[16] M Bökstedt, R R Bruner, S Lunøe-Nielsen, J Rognes, On cyclic fixed points of spectra, Math. Z. 276 (2014) 81 | DOI

[17] R R Bruner, J Rognes, The Adams spectral sequence for topological modular forms, 253, Amer. Math. Soc. (2021) | DOI

[18] R R Bruner, J P May, J E Mcclure, M Steinberger, H∞ ring spectra and their applications, 1176, Springer (1986) | DOI

[19] R Burklund, T M Schlank, A Yuan, The chromatic Nullstellensatz, preprint (2022)

[20] S G Chadwick, M A Mandell, En genera, Geom. Topol. 19 (2015) 3193 | DOI

[21] D Clausen, A Mathew, N Naumann, J Noel, Descent and vanishing in chromatic algebraic K–theory via group actions, Ann. Sci. Éc. Norm. Supér. 57 (2024) 1135

[22] R L Cohen, Odd primary infinite families in stable homotopy theory, 242, Amer. Math. Soc. (1981) | DOI

[23] F R Cohen, T J Lada, J P May, The homology of iterated loop spaces, 533, Springer (1976) | DOI

[24] F R Cohen, M E Mahowald, R J Milgram, The stable decomposition for the double loop space of a sphere, from: "Algebraic and geometric topology, II" (editor R J Milgram), Proc. Sympos. Pure Math. 32, Amer. Math. Soc. (1978) 225

[25] B I Dundas, Relative K-theory and topological cyclic homology, Acta Math. 179 (1997) 223 | DOI

[26] G Dunn, Tensor product of operads and iterated loop spaces, J. Pure Appl. Algebra 50 (1988) 237 | DOI

[27] J Hahn, D Wilson, Redshift and multiplication for truncated Brown–Peterson spectra, Ann. of Math. 196 (2022) 1277 | DOI

[28] J Hahn, A Raksit, D Wilson, A motivic filtration on the topological cyclic homology of commutative ring spectra, preprint (2022)

[29] A Hedenlund, J Rognes, A multiplicative Tate spectral sequence for compact Lie group actions, 1468, Amer. Math. Soc. (2024) | DOI

[30] L Hesselholt, Topological Hochschild homology and the Hasse–Weil zeta function, from: "An alpine bouquet of algebraic topology" (editors C Ausoni, K Hess, B Johnson, I Moerdijk, J Scherer), Contemp. Math. 708, Amer. Math. Soc. (2018) 157 | DOI

[31] L Hesselholt, I Madsen, On the K-theory of finite algebras over Witt vectors of perfect fields, Topology 36 (1997) 29 | DOI

[32] L Hesselholt, I Madsen, On the K-theory of local fields, Ann. of Math. 158 (2003) 1 | DOI

[33] M Land, A Mathew, L Meier, G Tamme, Purity in chromatically localized algebraic K–theory, J. Amer. Math. Soc. 37 (2024) 1011 | DOI

[34] T Lawson, N Naumann, Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2, Int. Math. Res. Not. 2014 (2014) 2773 | DOI

[35] L G Lewis Jr., J P May, M Steinberger, J E Mcclure, Equivariant stable homotopy theory, 1213, Springer (1986) | DOI

[36] A Liulevicius, The factorization of cyclic reduced powers by secondary cohomology operations, 42, Amer. Math. Soc. (1962) 112 | DOI

[37] S Lunøe-Nielsen, J Rognes, The topological Singer construction, Doc. Math. 17 (2012) 861 | DOI

[38] M Mahowald, C Rezk, Brown–Comenetz duality and the Adams spectral sequence, Amer. J. Math. 121 (1999) 1153 | DOI

[39] M A Mandell, J P May, Equivariant orthogonal spectra and S-modules, 755, Amer. Math. Soc. (2002) | DOI

[40] J P May, A general algebraic approach to Steenrod operations, from: "The Steenrod algebra and its applications" (editor F P Peterson), Lecture Notes in Math. 168, Springer (1970) 153 | DOI

[41] J E Mcclure, R E Staffeldt, On the topological Hochschild homology of bu, I, Amer. J. Math. 115 (1993) 1 | DOI

[42] J Milnor, The Steenrod algebra and its dual, Ann. of Math. 67 (1958) 150 | DOI

[43] T Nikolaus, P Scholze, On topological cyclic homology, Acta Math. 221 (2018) 203 | DOI

[44] D C Ravenel, Complex cobordism and stable homotopy groups of spheres, Amer. Math. Soc. (2004)

[45] J Rognes, Trace maps from the algebraic K-theory of the integers (after Marcel Bökstedt), J. Pure Appl. Algebra 125 (1998) 277 | DOI

[46] J Rognes, Topological cyclic homology of the integers at two, J. Pure Appl. Algebra 134 (1999) 219 | DOI

[47] J Rognes, Algebraic K-theory of finitely presented ring spectra, lecture notes (2000)

[48] J Rognes, Algebraic K-theory of strict ring spectra, from: "Proceedings of the International Congress of Mathematicians, II" (editors S Y Jang, Y R Kim, D W Lee, I Ye), Kyung Moon Sa (2014) 1259

[49] J Rognes, The circle action on topological Hochschild homology of complex cobordism and the Brown–Peterson spectrum, J. Topol. 13 (2020) 939 | DOI

[50] L Smith, On realizing complex bordism modules: applications to the stable homotopy of spheres, Amer. J. Math. 92 (1970) 793 | DOI

[51] A A Suslin, On the K-theory of local fields, J. Pure Appl. Algebra 34 (1984) 301 | DOI

[52] H Toda, Extended p-th powers of complexes and applications to homotopy theory, Proc. Japan Acad. 44 (1968) 198

[53] H Toda, On spectra realizing exterior parts of the Steenrod algebra, Topology 10 (1971) 53 | DOI

[54] S Tsalidis, Topological Hochschild homology and the homotopy descent problem, Topology 37 (1998) 913 | DOI

[55] V Voevodsky, On motivic cohomology with Z∕l-coefficients, Ann. of Math. 174 (2011) 401 | DOI

[56] S I Yanagida, Z I Yosimura, Ring spectra with coefficients in V (1) and V (2), I, Japan. J. Math. 3 (1977) 191 | DOI

[57] Z I Yosimura, Ring spectra with coefficients in V (1) and V (2), II, Japan. J. Math. 3 (1977) 297 | DOI

[58] A Yuan, Examples of chromatic redshift in algebraic K-theory, J. Eur. Math. Soc. (2024) | DOI

[59] R Zahler, The Adams–Novikov spectral sequence for the spheres, Ann. of Math. 96 (1972) 480 | DOI