The first cohomology group of the generalized Morava stabilizer algebra

Authors:
Hirofumi Nakai and Douglas C. Ravenel

Journal:
Proc. Amer. Math. Soc. **131** (2003), 1629-1639

MSC (2000):
Primary 55P42, 55T15; Secondary 14L05, 20Jxx

Published electronically:
September 19, 2002

MathSciNet review:
1950296

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: There exists a -local spectrum with = . Its Adams-Novikov -term is isomorphic to

where

In this paper we determine the groups

for all . Its rank ranges from to depending on the value of .

**[Ich]**I. Ichigi.

The chromatic groups at the prime two.

To appear in Mem. Fac. Kochi Univ. (Math.).**[IK00]**Ippei Ichigi and Katsumi Shimomura,*The chromatic 𝐸₁-term 𝐸𝑥𝑡⁰(𝑣⁻¹₃𝐵𝑃_{∗}/(3,𝑣₁,𝑣^{∞}₂)[𝑡₁])*, Mem. Fac. Sci. Kochi Univ. Ser. A Math.**21**(2000), 63–71. MR**1744540****[INR]**I. Ichigi, H. Nakai, and D. C. Ravenel.

The chromatic Ext groups .*Trans. Amer. Math. Soc*., 354:3789-3813, 2002.**[KS93]**N. Kodama and K. Shimomura.

On the homotopy groups of a spectrum related to Ravenel's spectra .*J. Fac. Educ. Tottori Univ. (Nat. Sci.)*, 42:17-30, 1993.**[KS01]**Y. Kamiya and K. Shimomura.

The homotopy groups .*Hiroshima Mathematical Journal*, 31:391-408, 2001.**[May66]**J. P. May,*The cohomology of restricted Lie algebras and of Hopf algebras*, J. Algebra**3**(1966), 123–146. MR**0193126****[MM65]**John W. Milnor and John C. Moore,*On the structure of Hopf algebras*, Ann. of Math. (2)**81**(1965), 211–264. MR**0174052****[MR77]**Haynes R. Miller and Douglas C. Ravenel,*Morava stabilizer algebras and the localization of Novikov’s 𝐸₂-term*, Duke Math. J.**44**(1977), no. 2, 433–447. MR**0458410****[MRW77]**Haynes R. Miller, Douglas C. Ravenel, and W. Stephen Wilson,*Periodic phenomena in the Adams-Novikov spectral sequence*, Ann. of Math. (2)**106**(1977), no. 3, 469–516. MR**0458423****[MS93a]**Mark Mahowald and Katsumi Shimomura,*The Adams-Novikov spectral sequence for the 𝐿₂ localization of a 𝑣₂ spectrum*, Algebraic topology (Oaxtepec, 1991) Contemp. Math., vol. 146, Amer. Math. Soc., Providence, RI, 1993, pp. 237–250. MR**1224918**, 10.1090/conm/146/01226**[MS93b]**H. Mitsui and K. Shimomura.

The Ext groups .*J. Fac. Educ. Tottori Univ. (Nat. Sci.)*, 42:85-101, 1993.**[NRa]**H. Nakai and D. C. Ravenel.

The method of infinite descent in stable homotopy theory II.

To appear.**[NRb]**H. Nakai and D. C. Ravenel.

The structure of the general chromatic -term and .

To appear in Osaka J. Math.**[NY]**H. Nakai and D. Yoritomi.

The structure of the general chromatic -term for .

To appear.**[Rav86]**Douglas C. Ravenel,*Complex cobordism and stable homotopy groups of spheres*, Pure and Applied Mathematics, vol. 121, Academic Press, Inc., Orlando, FL, 1986. MR**860042****[Rav00]**Douglas C. Ravenel,*The microstable Adams-Novikov spectral sequence*, Une dégustation topologique [Topological morsels]: homotopy theory in the Swiss Alps (Arolla, 1999) Contemp. Math., vol. 265, Amer. Math. Soc., Providence, RI, 2000, pp. 193–209. MR**1803959**, 10.1090/conm/265/04250**[Rav02]**D. C. Ravenel.

The method of infinite descent in stable homotopy theory I.

In D. M. Davis, editor,*Recent Progress in Homotopy Theory*, volume 293 of*Contemporary Mathematics*, pages 251-284, Providence, Rhode Island, 2002. American Mathematical Society.**[Shia]**Katsumi Shimomura,*The homotopy groups of the 𝐿₂-localized Mahowald spectrum 𝑋⟨1⟩*, Forum Math.**7**(1995), no. 6, 685–707. MR**1359422**, 10.1515/form.1995.7.685**[Shic]**K. Shimomura.

The homotopy groups . Recent progress in homotopy theory (Baltimore, MD, 2000), 285-297, Contemp. Math., 293, Amer. Math. Soc., Providence, RI, 2002.**[Shi95]**K. Shimomura.

Chromatic -terms - up to April 1995.*J. Fac. Educ. Tottori Univ. (Nat. Sci.)*, 44:1-6, 1995.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
55P42,
55T15,
14L05,
20Jxx

Retrieve articles in all journals with MSC (2000): 55P42, 55T15, 14L05, 20Jxx

Additional Information

**Hirofumi Nakai**

Affiliation:
Oshima National College of Maritime Technology, 1091-1 komatsu Oshima-cho Oshima-gun, Yamaguchi 742-2193, Japan

Email:
nakai@c.oshima-k.ac.jp

**Douglas C. Ravenel**

Affiliation:
Department of Mathematics, University of Rochester, Rochester, New York 14627

Email:
drav@math.rochester.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-02-06718-7

Received by editor(s):
June 14, 2001

Received by editor(s) in revised form:
December 19, 2001

Published electronically:
September 19, 2002

Additional Notes:
The second author acknowledges support from NSF grant DMS-9802516

Communicated by:
Paul Goerss

Article copyright:
© Copyright 2002
American Mathematical Society