THE RO(G)-GRADED EQUIVARIANT ORDINARY COHOMOLOGY OF COMPLEX PROJECTIVE SPACES WITH LINEAR 2[/p ACTIONS

INTRODUCTION. If X is a CW complex with cells only in even dimensions and R is a ring, then, by an elementary result in cellular cohomology theory, the ordinary eohomology H*(X;R) of X with R coefficients is a free, 7/-graded R-module. Since this result is quite useful in the study of well-behaved complex manifolds like projective spaces or Grassmannians, it would be nice to be able to generalize it to equivariant ordinary eohomology. The result does generalize in the following sense. Let G be a finite group, X be a G-CW complex (in the sense of [MAT, LMSM]), and R be a ring-valued eontravariant coefficient system JILL]. Then the G-equivariant ordinary Bredon cohomology H*(X; R) of X with R coefficients may be regarded as a coefficient system. If the cells of X are all even dimensional, then H*(X;R) is a free module over R in the sense appropriate to coefficient systems. Unfortunately, this theorem does not apply to complex projective spaces or complex Grassmannians with any reasonable nontrivial G-action because these spaces do not have the right kind of G-CW structure. In fact, if G is ~/p, for any prime p, and r / is a nontrivial irreducible complex G-representation, then the theorem does not apply to S ~, the onepoint compactification of r 1. Moreover, the 2~-graded Bredon cohomology of S n with coefficients in the Burnside ring coefficient system is quite obviously not free over the coefficient system.