Cohomology of uniformly powerful 𝑝-groups

Browder, William; Pakianathan, Jonathan

doi:10.1090/S0002-9947-99-02470-8

Cohomology of uniformly powerful $p$-groups
HTML articles powered by AMS MathViewer

by William Browder and Jonathan Pakianathan PDF

Trans. Amer. Math. Soc. 352 (2000), 2659-2688 Request permission

Abstract:

In this paper we will study the cohomology of a family of $p$-groups associated to $\mathbb {F}_p$-Lie algebras. More precisely, we study a category $\mathbf {BGrp}$ of $p$-groups which will be equivalent to the category of $\mathbb {F}_p$-bracket algebras (Lie algebras minus the Jacobi identity). We then show that for a group $G$ in this category, its $\mathbb {F}_p$-cohomology is that of an elementary abelian $p$-group if and only if it is associated to a Lie algebra. We then proceed to study the exponent of $H^*(G ;\mathbb {Z})$ in the case that $G$ is associated to a Lie algebra $\mathfrak {L}$. To do this, we use the Bockstein spectral sequence and derive a formula that gives $B_2^*$ in terms of the Lie algebra cohomologies of $\mathfrak {L}$. We then expand some of these results to a wider category of $p$-groups. In particular, we calculate the cohomology of the $p$-groups $\Gamma _{n,k}$ which are defined to be the kernel of the mod $p$ reduction $GL_n(\mathbb {Z}/p^{k+1}\mathbb {Z}) \overset {mod}{\longrightarrow } GL_n(\mathbb {F}_p).$

References

Alejandro Adem, Cohomological exponents of $\textbf {Z}G$-lattices, J. Pure Appl. Algebra 58 (1989), no. 1, 1–5. MR 996170, DOI 10.1016/0022-4049(89)90048-0
D. J. Benson, Representations and cohomology. II, Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, Cambridge, 1991. Cohomology of groups and modules. MR 1156302
D. J. Benson and Jon F. Carlson, The cohomology of extraspecial groups, Bull. London Math. Soc. 24 (1992), no. 3, 209–235. MR 1157256, DOI 10.1112/blms/24.3.209
D. J. Benson and Jon F. Carlson, Erratum to: “The cohomology of extraspecial groups” [Bull. London Math. Soc. 24 (1992), no. 3, 209–235; MR1157256 (93b:20087)], Bull. London Math. Soc. 25 (1993), no. 5, 498. MR 1233415, DOI 10.1112/blms/25.5.498
Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 1–3, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1975 edition. MR 979493
W. Browder, J. Pakianathan, Lifting Lie algebras over the residue field of a discrete valuation ring, to appear, Journal of Pure and Applied Algebra.
Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1994. Corrected reprint of the 1982 original. MR 1324339
J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic pro-$p$-groups, London Mathematical Society Lecture Note Series, vol. 157, Cambridge University Press, Cambridge, 1991. MR 1152800
Morikuni Goto and Frank D. Grosshans, Semisimple Lie algebras, Lecture Notes in Pure and Applied Mathematics, Vol. 38, Marcel Dekker, Inc., New York-Basel, 1978. MR 0573070
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
J. Pakianathan, On the cohomology of a family of p-groups associated to Lie algebras, Princeton University thesis, 1997.
Helmut Strade and Rolf Farnsteiner, Modular Lie algebras and their representations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 116, Marcel Dekker, Inc., New York, 1988. MR 929682
T. Weigel, Combinatorial properties of p-central groups, preprint.
T. Weigel, P-central groups and Poincar$\acute {e}$ duality, Trans. Amer. Math. Soc., to appear.
T. Weigel, Exp and log functors for the categories of powerful p-central groups and Lie algebras, Habilitationsschrift, Freiburg, 1994.
T. Weigel, A note on the Ado-Iwasawa theorem, in preparation.