The RO(C4) cohomology of the infinite real projective space

Following the Hu-Kriz method of computing the C 2 genuine dual Steenrod algebra ?.H F 2 ^H F 2 / C 2 , we calculate the C 4 -equivariant Bredon cohomology of the classifying space RP 1 D B C 4 † 2 as an RO.C 4 / graded Green-functor.We prove that as a module over the homology of a point (which we also compute), this cohomology is not flat.As a result, it can't be used as a test module for obtaining generators in ?.H F 2 ^H F 2 / C 4 as Hu and Kriz use it in the C 2 case.Their argument for the Borel equivariant dual Steenrod algebra does generalize, however, and we give a complete description of ?.H F 2 ^H F 2 / hC 2 n for any n 2. 55N91, 55P91