Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Localization of equivariant cohomology rings

Author: J. Duflot
Journal: Trans. Amer. Math. Soc. 284 (1984), 91-105
MSC: Primary 57S15; Secondary 20J06, 55N91
Erratum: Trans. Amer. Math. Soc. 290 (1985), 857-858.
MathSciNet review: 742413
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

$\displaystyle {H^{\ast} }{(BG,{\mathbf{Z}}/p{\mathbf{Z}})_{{\mathfrak{P}_A}}} \... ...{H^{\ast} }(B{C_G}(A),{\mathbf{Z}}/p{\mathbf{Z}})_{{\mathfrak{P}_A}}^{{W_G}(A)}$

. 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 [Enhancements On Off] (What's this?)

  • [B] A. Borel et al., Seminar on compact transformation groups, Ann. of Math. Studies, vol. 46, Princeton Univ. Press, Princeton, N. J., 1960. MR 0116341 (22:7129)
  • [Br] G. Bredon. Introduction to compact transformation groups, Academic Press, New York, 1972. MR 0413144 (54:1265)
  • [C-E] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N. J., 1954. MR 1731415 (2000h:18022)
  • [D1] J. Duflot, Depth and equivariant cohomology, Comment. Math. Helv. 56 (1981), 627-637. MR 656216 (83g:57029)
  • [H] Wu-Yi Hsiang, Cohomology theory of topological transformation groups, Springer-Verlag, New York, 1975. MR 0423384 (54:11363)
  • [M] H. Matsumura, Commutative algebra, Benjamin, New York, 1970. MR 0266911 (42:1813)
  • [Mui] H. Mui, Modular invariant theory and cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math 22 (1975), 319-369. MR 0422451 (54:10440)
  • [Q1, Q2] D. Quillen, The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. (2) 94 (1971), 549-602. MR 0298694 (45:7743)
  • [T] T. tom Dieck, Transformation groups and representation theory, Lecture Notes in Math., vol. 766, Springer-Verlag, Berlin and New York, 1979. MR 551743 (82c:57025)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57S15, 20J06, 55N91

Retrieve articles in all journals with MSC: 57S15, 20J06, 55N91

Additional Information

Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society