Localization of equivariant cohomology rings
HTML articles powered by AMS MathViewer
- by J. Duflot PDF
- Trans. Amer. Math. Soc. 284 (1984), 91-105 Request permission
Erratum: Trans. Amer. Math. Soc. 290 (1985), 857-858.
Abstract:
The main result of this paper is the "calculation" of the Borel equivariant cohomology ring ${H^{\ast } }(EG \times _G X,{\mathbf {Z}}/p{\mathbf {Z}})$ localized at one of its minimal prime ideals. In case $X$ is a point, the work of Quillen shows that the minimal primes ${\mathfrak {P}_A}$ are parameterized by the maximal elementary abelian $p$-subgroups $A$ of $G$ and the result is \[ {H^{\ast } }{(BG,{\mathbf {Z}}/p{\mathbf {Z}})_{{\mathfrak {P}_A}}} \cong {H^{\ast } }(B{C_G}(A),{\mathbf {Z}}/p{\mathbf {Z}})_{{\mathfrak {P}_A}}^{{W_G}(A)}\]. Here, ${C_G}(A)$ is the centralizer of $A$ in $G$, and ${W_G}(A) = {N_G}(A)/{C_G}(A)$, where ${N_G}(A)$ is the normalizer of $A$ in $G$. An example is included.References
- Armand Borel, Seminar on transformation groups, Annals of Mathematics Studies, No. 46, Princeton University Press, Princeton, N.J., 1960. With contributions by G. Bredon, E. E. Floyd, D. Montgomery, R. Palais. MR 0116341
- Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
- Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1999. With an appendix by David A. Buchsbaum; Reprint of the 1956 original. MR 1731415
- J. Duflot, Depth and equivariant cohomology, Comment. Math. Helv. 56 (1981), no. 4, 627–637. MR 656216, DOI 10.1007/BF02566231
- Wu-yi Hsiang, Cohomology theory of topological transformation groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 85, Springer-Verlag, New York-Heidelberg, 1975. MR 0423384
- Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911
- Huỳnh Mui, Modular invariant theory and cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), no. 3, 319–369. MR 422451
- 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
- Tammo tom Dieck, Transformation groups and representation theory, Lecture Notes in Mathematics, vol. 766, Springer, Berlin, 1979. MR 551743
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 284 (1984), 91-105
- MSC: Primary 57S15; Secondary 20J06, 55N91
- DOI: https://doi.org/10.1090/S0002-9947-1984-0742413-X
- MathSciNet review: 742413