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The units of a ring spectrum and a logarithmic cohomology operation. (English) Zbl 1106.55002
Let $$R$$ be a spectrum with additional structure making it a commutative $$S$$-algebra in the sense of A. K. Elmendorf, I. Kriz, M. A. Mandell and J. P. May [Rings, Modules and Algebras. Mathematical Surveys and Monographs. 47. (Providence), RI: American Mathematical Society (AMS). (1997; Zbl 0894.55001)], $$\text{GL}_1(R)$$ the grouplike $$E_\infty$$-space determined by the pullback $\begin{tikzcd}GL_1(R) \rar\dar & \Omega^\infty R\dar\\ (\pi_0 R)^\times \rar & \pi_0 R \end{tikzcd}$ and $$\text{gl}_1(R)$$ the $$(-1)$$ connected spectrum with $$\text{GL}_1(R)$$ as $$0$$-space. The cohomology theory associated to $$\text{gl}_1(R)$$ has the property that $$\text{gl}_1(R)^0 (X) = (R^0(X))^\times$$ (and so is referred to as the units of the spectrum $$R$$). For $$n \geq 1$$ and $$p$$ a prime the Bousfield-Kuhn construction [A. K. Bousfield, Pac. J. Math 129, 1–31 (1987; Zbl 0664.55006), N. Kuhn, Algebraic topology, Proc. Int. Conf., Arcata/Calif. 1986, Lect. Notes Math. 1370, 243–257 (1989; Zbl 0692.55005)] determines a natural map $$\ell_{n,p}: \text{gl}_1(R) \to L_{K(n)}(R)$$ where $$L_{K(n)}$$ denotes localization with respect to $$K(n)$$, the $$n$$-th Morava $$K$$-theory. The main result of the paper under review is a formula for the map $\ell_{n,p}: (R^0 X)^\times \to (L_{K(n)} R)^0 (X)$ in the case that $$R$$ is equal to $$E$$, a Morava $$E$$-theory associated to a height $$n \geq 1$$ formal group law over a perfect field of characterisitc $$p$$. For these theories power operations can be constructed [M. Ando, Duke Math. J. 79, 423–485 (1995; Zbl 0862.55004)] by showing that for every finite subgroup, $$A$$, of $$\Lambda^* = ({\mathbb Q}_p / {\mathbb Z}_p)^n$$ there is associated a natural ring homomorphism $\Psi_A: E^0 (X) \to D \otimes_{E_0} E^0 (X).$ The author shows that for $$x \in (E^0 X)^\times$$ $\ell_{n,p}(x) = \sum^\infty_{k=1} (-1)^{k-1} \frac{p^{k-1}}{k} M(x)^k$ where $$M: E^0(X) \to E^0(X)$$ is the unique cohomology operation such that $1+p M(x) = \prod^n_{j=0} \Biggl( \prod_{_{\substack{ A \subseteq \Lambda^* [p] \\ | A| = p^j }}} \Psi_A (x) \Biggr)^{(-1)^j (j-1)(j-2)/2}.$ Here $$\Lambda^*[p]$$ denotes the kernel of multiplication by $$p$$ on $$\Lambda^*$$. The author also expresses this formula in terms of certain additive $$E$$-operations which he calls Hecke operators and notes the similarity with expressions arising in the theory of $$L$$-functions.

##### MSC:
 55N22 Bordism and cobordism theories and formal group laws in algebraic topology 55P43 Spectra with additional structure ($$E_\infty$$, $$A_\infty$$, ring spectra, etc.) 55S05 Primary cohomology operations in algebraic topology 55S25 $$K$$-theory operations and generalized cohomology operations in algebraic topology 55P47 Infinite loop spaces 55P60 Localization and completion in homotopy theory 55N34 Elliptic cohomology 11F25 Hecke-Petersson operators, differential operators (one variable)
##### Keywords:
cohomology; operation; spectrum; Morava E-theory
Full Text:
##### References:
 [1] Matthew Ando, Michael J. Hopkins, and Neil P. Strickland, The sigma orientation is an \?_\infty map, Amer. J. Math. 126 (2004), no. 2, 247 – 334. · Zbl 1071.55003 [2] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. · Zbl 0175.03601 [3] Matthew Ando, Isogenies of formal group laws and power operations in the cohomology theories \?_\?, Duke Math. J. 79 (1995), no. 2, 423 – 485. · Zbl 0862.55004 · doi:10.1215/S0012-7094-95-07911-3 · doi.org [4] M. F. Atiyah and G. B. Segal, Exponential isomorphisms for \?-rings, Quart. J. Math. Oxford Ser. (2) 22 (1971), 371 – 378. · Zbl 0226.13008 · doi:10.1093/qmath/22.3.371 · doi.org [5] A. K. Bousfield and E. M. Friedlander, Homotopy theory of \Gamma -spaces, spectra, and bisimplicial sets, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 80 – 130. · Zbl 0405.55021 [6] Martin Bendersky and John R. Hunton, On the coalgebraic ring and Bousfield-Kan spectral sequence for a Landweber exact spectrum, Proc. Edinb. Math. Soc. (2) 47 (2004), no. 3, 513 – 532. · Zbl 1068.55010 · doi:10.1017/S0013091503000518 · doi.org [7] R. R. Bruner, J. P. May, J. E. McClure, and M. Steinberger, \?_\infty ring spectra and their applications, Lecture Notes in Mathematics, vol. 1176, Springer-Verlag, Berlin, 1986. · Zbl 0585.55016 [8] A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), no. 4, 257 – 281. · Zbl 0417.55007 · doi:10.1016/0040-9383(79)90018-1 · doi.org [9] A. K. Bousfield, Uniqueness of infinite deloopings for \?-theoretic spaces, Pacific J. Math. 129 (1987), no. 1, 1 – 31. · Zbl 0664.55006 [10] A. K. Bousfield, On the telescopic homotopy theory of spaces, Trans. Amer. Math. Soc. 353 (2001), no. 6, 2391 – 2426. · Zbl 0971.55016 [11] Albrecht Dold, Decomposition theorems for \?(\?)-complexes, Ann. of Math. (2) 75 (1962), 8 – 16. · Zbl 0125.01201 · doi:10.2307/1970415 · doi.org [12] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997. With an appendix by M. Cole. · Zbl 0894.55001 [13] P. G. Goerss and M. J. Hopkins, Moduli spaces of commutative ring spectra, Structured ring spectra, London Math. Soc. Lecture Note Ser., vol. 315, Cambridge Univ. Press, Cambridge, 2004, pp. 151-200. · Zbl 1086.55006 [14] Paul G. Goerss and John F. Jardine, Simplicial homotopy theory, Progress in Mathematics, vol. 174, Birkhäuser Verlag, Basel, 1999. · Zbl 0949.55001 [15] David Gluck, Idempotent formula for the Burnside algebra with applications to the \?-subgroup simplicial complex, Illinois J. Math. 25 (1981), no. 1, 63 – 67. · Zbl 0424.16007 [16] Michael J. Hopkins, Nicholas J. Kuhn, and Douglas C. Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000), no. 3, 553 – 594. · Zbl 1007.55004 [17] Luke Hodgkin, The \?-theory of some wellknown spaces. I. \?\?$$^{0}$$, Topology 11 (1972), 371 – 375. · Zbl 0246.55003 · doi:10.1016/0040-9383(72)90032-8 · doi.org [18] M. J. Hopkins, Algebraic topology and modular forms, Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 291 – 317. · Zbl 1031.55007 [19] Mark Hovey and Neil P. Strickland, Morava \?-theories and localisation, Mem. Amer. Math. Soc. 139 (1999), no. 666, viii+100. · Zbl 0929.55010 · doi:10.1090/memo/0666 · doi.org [20] Mark Hovey, Brooke Shipley, and Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), no. 1, 149 – 208. · Zbl 0931.55006 [21] Takuji Kashiwabara, Hopf rings and unstable operations, J. Pure Appl. Algebra 94 (1994), no. 2, 183 – 193. · Zbl 0824.55008 · doi:10.1016/0022-4049(94)90032-9 · doi.org [22] Nicholas J. Kuhn, Localization of André-Quillen-Goodwillie towers, and the periodic homology of infinite loopspaces, preprint. · Zbl 1103.55007 [23] Nicholas J. Kuhn, Morava \?-theories and infinite loop spaces, Algebraic topology (Arcata, CA, 1986) Lecture Notes in Math., vol. 1370, Springer, Berlin, 1989, pp. 243 – 257. · doi:10.1007/BFb0085232 · doi.org [24] L. G. Lewis Jr., J. P. May, M. Steinberger, and J. E. McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure. · Zbl 0611.55001 [25] Jonathan Lubin and John Tate, Formal moduli for one-parameter formal Lie groups, Bull. Soc. Math. France 94 (1966), 49 – 59. · Zbl 0156.04105 [26] J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, Berlin-New York, 1973. J. P. May, The geometry of iterated loop spaces, Springer-Verlag, Berlin-New York, 1972. Lectures Notes in Mathematics, Vol. 271. · Zbl 0285.55012 [27] J. Peter May, \?_\infty ring spaces and \?_\infty ring spectra, Lecture Notes in Mathematics, Vol. 577, Springer-Verlag, Berlin-New York, 1977. With contributions by Frank Quinn, Nigel Ray, and Jørgen Tornehave. · Zbl 0345.55007 [28] J. P. May, Multiplicative infinite loop space theory, J. Pure Appl. Algebra 26 (1982), no. 1, 1 – 69. · Zbl 0532.55013 · doi:10.1016/0022-4049(82)90029-9 · doi.org [29] M. A. Mandell and J. P. May, Equivariant orthogonal spectra and \?-modules, Mem. Amer. Math. Soc. 159 (2002), no. 755, x+108. · Zbl 1025.55002 · doi:10.1090/memo/0755 · doi.org [30] Birgit Richter and Alan Robinson, Gamma homology of group algebras and of polynomial algebras, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic \?-theory, Contemp. Math., vol. 346, Amer. Math. Soc., Providence, RI, 2004, pp. 453 – 461. · Zbl 1072.55005 · doi:10.1090/conm/346/06298 · doi.org [31] Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Kanô Memorial Lectures, No. 1. · Zbl 0221.10029 [32] Neil P. Strickland and Paul R. Turner, Rational Morava \?-theory and \?\?$$^{0}$$, Topology 36 (1997), no. 1, 137 – 151. · Zbl 0861.55007 · doi:10.1016/0040-9383(95)00073-9 · doi.org [33] Neil P. Strickland, Finite subgroups of formal groups, J. Pure Appl. Algebra 121 (1997), no. 2, 161 – 208. · Zbl 0916.14025 · doi:10.1016/S0022-4049(96)00113-2 · doi.org [34] N. P. Strickland, Morava \?-theory of symmetric groups, Topology 37 (1998), no. 4, 757 – 779. · Zbl 0912.55012 · doi:10.1016/S0040-9383(97)00054-2 · doi.org [35] Richard Woolfson, Hyper-\Gamma -spaces and hyperspectra, Quart. J. Math. Oxford Ser. (2) 30 (1979), no. 118, 229 – 255. · Zbl 0411.55006 · doi:10.1093/qmath/30.2.229 · doi.org
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