## Small subalgebras of Steenrod and Morava stabilizer algebras

HTML articles powered by AMS MathViewer

- 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.Β 1, 61β95. 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.Β 2, 435β454. 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.Β 3, 293β307. 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.Β 1-3, 155β168. 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), 665β676. - Daniel Quillen,
*The spectrum of an equivariant cohomology ring. I, II*, Ann. of Math. (2)**94**(1971), 549β572; ibid. (2) 94 (1971), 573β602. MR**298694**, DOI 10.2307/1970770 - Douglas C. Ravenel,
*The structure of Morava stabilizer algebras*, Invent. Math.**37**(1976), no.Β 2, 109β120. MR**420619**, DOI 10.1007/BF01418965 - Douglas C. Ravenel,
*The cohomology of the Morava stabilizer algebras*, Math. Z.**152**(1977), no.Β 3, 287β297. 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), 103β111. MR**215896** - Michishige Tezuka,
*Cohomology of unipotent algebraic and finite groups and the Steenrod algebra*, Math. Z.**216**(1994), no.Β 1, 45β67. MR**1273465**, DOI 10.1007/BF02572308 - Nobuaki Yagita,
*Frobenius operations and cohomology of $\textrm {GL}_3(\textbf {F}_q)$*, Comm. Algebra**16**(1988), no.Β 5, 989β1016. MR**926333**, DOI 10.1080/00927878808823614

## Additional Information

**N. Yagita**- Affiliation: Faculty of Education, Ibaraki University, Mito, Ibaraki, Japan
- MR Author ID: 185110
- Email: yagita@mito.ipc.ibaraki.ac.jp
- Received by editor(s): January 9, 1995
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**350**(1998), 3021-3041 - MSC (1991): Primary 55N22; Secondary 57R77
- DOI: https://doi.org/10.1090/S0002-9947-98-02226-0
- MathSciNet review: 1475699