Small subalgebras of Steenrod and Morava stabilizer algebras

Yagita, N.

doi:10.1090/S0002-9947-98-02226-0

Small subalgebras of Steenrod and Morava stabilizer algebras
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by N. Yagita PDF

Trans. Amer. Math. Soc. 350 (1998), 3021-3041 Request permission

Abstract:

Let $P(j)$ (resp. $S(n)_{(j)})$ be the subalgebra of the Steenrod algebra $\mathcal {A}_p$ (resp. $n$th Morava stabilizer algebra) generated by reduced powers $\mathcal {P}^{p^i}$, $0\le i\le j$ (resp. $t_i$, $1\le i\le j)$. In this paper we identify the dual $P(j-1)^*$ of $P(j-1)$ (resp. $S(n)_{(j)}$, for $j\le n)$ with some Frobenius kernel (resp. $F_{p^n}$-points) of a unipotent subgroup $G(j+1)$ of the general linear algebraic group $GL_{j+1}$. Using these facts, we get the additive structure of $H^*(P(1))=\operatorname {Ext}_{P(1)}(Z/p,Z/p)$ for odd primes.

References

Leonard Evens, The cohomology of groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1991. Oxford Science Publications. MR 1144017
V. Gorbounov, S. Siegel, and P. Symonds, The cohomology of the Morava stabilizer group $S_2$ at the prime $3$, Preprint 1994.
Masaharu Kaneda, Nobuo Shimada, Michishige Tezuka, and Nobuaki Yagita, Cohomology of infinitesimal algebraic groups, Math. Z. 205 (1990), no. 1, 61–95. MR 1069485, DOI 10.1007/BF02571225
Masaharu Kaneda, Nobuo Shimada, Michishige Tezuka, and Nobuaki Yagita, Representations of the Steenrod algebra, J. Algebra 155 (1993), no. 2, 435–454. MR 1212238, DOI 10.1006/jabr.1993.1053
H-W. Henn, On the $\operatorname {mod} p$ cohomology of profinite groups of positive $p$ rank, Preprint, 1994.
Haynes Miller and Clarence Wilkerson, Vanishing lines for modules over the Steenrod algebra, J. Pure Appl. Algebra 22 (1981), no. 3, 293–307. MR 629336, DOI 10.1016/0022-4049(81)90104-3
I. Leary, The cohomology of certain finite groups, Thesis, Cambridge Univ., 1990.
Ian Leary, A differential in the Lyndon-Hochschild-Serre spectral sequence, J. Pure Appl. Algebra 88 (1993), no. 1-3, 155–168. MR 1233320, DOI 10.1016/0022-4049(93)90019-P
Arunas Liulevicius, The factorization of cyclic reduced powers by secondary cohomology operations, Mem. Amer. Math. Soc. 42 (1962), 112. MR 182001
C. Peterson and N. Yagita, Rational cohomology of Witt groups, Math. Z. 224 (1997), 665–676.
Daniel Quillen, The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. (2) 94 (1971), 549–572; ibid. (2) 94 (1971), 573–602. MR 298694, DOI 10.2307/1970770
Douglas C. Ravenel, The structure of Morava stabilizer algebras, Invent. Math. 37 (1976), no. 2, 109–120. MR 420619, DOI 10.1007/BF01418965
Douglas C. Ravenel, The cohomology of the Morava stabilizer algebras, Math. Z. 152 (1977), no. 3, 287–297. MR 431168, DOI 10.1007/BF01488970
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
Nobuo Shimada and Akira Iwai, On the cohomology of some Hopf algebras, Nagoya Math. J. 30 (1967), 103–111. MR 215896
Michishige Tezuka, Cohomology of unipotent algebraic and finite groups and the Steenrod algebra, Math. Z. 216 (1994), no. 1, 45–67. MR 1273465, DOI 10.1007/BF02572308
Nobuaki Yagita, Frobenius operations and cohomology of $\textrm {GL}_3(\textbf {F}_q)$, Comm. Algebra 16 (1988), no. 5, 989–1016. MR 926333, DOI 10.1080/00927878808823614