Equivariant Motivic Cohomology

The present paper will form part of the author’s PhD thesis, which will concern in part the practical computation of the motivic cohomology and the equivariant motivic cohomology of homogeneous varieties, such as Stiefel manifolds, Grassmanians, and spaces of matrices with prescribed rank conditions. Nothing proved here is particularly surprising, but it seems to the author that the spectral sequence of proposition 11 is likely to be of general interest, especially in the case where all schemes considered are smooth, in which case the isomorphism H(X;R) = CH(X, h)R holds and we have a spectral sequence in terms of the higher Chow groups. We establish in proposition 11 a spectral sequence computing the motivic cohomology of a homogeneous variety X = Y/G in terms of the motivic cohomology of Y and G. This is the analogue of a spectral sequence in classical algebraic topology that goes by several names, “Rothenberg-Steenrod” or “fiber-to-base EilenbergMoore”. When this sequence is applied in algebriac topology, because Y is a principal G-bundle over X, the map Y → X is a fibration, which is not the case in the A-homotopy theory. In classical topology, the fiber sequence can be continued by delooping to give another fiber sequence Y → EG×G Y ' X → BG, in which the space of interest is the total space, and so the computation may be carried out by the Serre spectral sequence. In the A-homotopy theory where we do not have the Serre spectral sequence at our disposal, the sequence established here is much more useful. In the case where a space Y has a non-free G-action, the sequence we construct still converges, to the cohomology of a homotopy type B(pt, G, Y) which depends functorially on Y. We offer this as a plausible definition of the G-equivariant homotopy type of Y. We are able to demonstrate that, at least in ideal cases, the equivariant motivic cohomology computed by our methods agrees with the equivariant motivic cohomology defined by [EG98]