Computation of cohomology of Morava $K$-Theory

Question

Let $G$ be an elementary abelian group, so that $G = (\mathbb{Z}/p)^k$ for some $k$. We can then compute the Morava $K$-theory of $BG = (BZ/p)^k$ pretty easily: $K(n)^*(BG) = K(n)^*[[x]]/[p](x) (x)_{K(n)^*} ... (x)_{K(n)^*} K(n)^*[[x]]/[p](x)$, the tensor being taken $k$ times. We know that $[p](x) = x^{p^n}$. What is this cohomology $K(n)^*(BG)$ as a representation of $GL_k(\mathbb{F}$_p)$? Maybe we can compute the topological cyclic homology via a slice spectral sequence computation.

Answer 1

Bjorn Schuster and I looked at this problem in our article `On the $GL(V)$-module structure of $K(n)^*(BV)$' Math Proc Cambridge Phil Soc vol 122 (1997) pp73--89. In particular, we show that in at least some cases it is not a permutation module. If you don't have access to the journal, there is a preprint that started off on Dave Benson's preprint server and was put on ArXiv in 2007: https://doi.org/10.48550/arXiv.0711.5016

Before our article, Nick Kuhn had shown that $K(n)^*(BV)$ always has the same Brauer character as a permutation module for $GL(V)$ in `Morava $K$-theories of some classifying spaces' Trans Amer Math Soc vol 304 (1987) 193--205, and this motivated our work.

Answer 2

Depending on exactly what you are trying to do, you may find it more useful to consider $E^0(BV)$, where $E$ is Morava $E$-theory and $V\simeq(\mathbb{Z}/p)^d$. (Here everything is $2$-periodic and concentrated in even degrees, so it is sufficient to consider degree zero.) For each subgroup $W\leq V$ we can consider $E^0(BW)$ as a quotient of $E^0(BV)$. For each character $\alpha\colon W\to S^1$ we have an Euler class $x_\alpha\in E^0(BW)$. We can let $r_W(t)$ be the remainder when $[p](t)$ is divided by $\prod_\alpha(t-x_\alpha)$, and let $D_W$ be the quotient of $E^0(BW)$ by the ideal generated by the coefficients of $r_W(t)$. This is called the classifying ring for level structures; it is a complete regular local noetherian ring and is a free module of finite rank over $E^0(\text{point})$ so it has very nice properties from the point of view of commutative algebra. The natural map $E^0(BV)\to\prod_WD_W$ is injective and becomes an isomorphism after tensoring with $\mathbb{Q}$. There is also a natural filtration of $E^0(BV)$ whose associated graded is $\prod_WD_W$. The action of $\text{Aut}(V)$ permutes the factors in this product in the obvious way. This still leaves an action of $\text{Aut}(W)$ on $D_W$. Although Galois theory does not always work well for group actions on rings that are not fields, in this particular case the behaviour is quite good. A lot of this is covered in my paper Finite subgroups of formal groups (published version, ungated version). There are more delicate results in the same direction in my paper Varieties and local cohomology for chromatic group cohomology rings (with John Greenlees) (published version, ungated version); I will not attempt to summarise them here.

As part of the same circle of ideas, the generalised character theory of Hopkins, Kuhn and Ravenel provides a ring $L$ (which is a free module over $\mathbb{Q}\otimes E^0(\text{point})$) and a natural isomorphism $L\otimes_{E^0(\text{point})}E^0(BV)\to\text{Map}(V^n,L)$. This makes it easy to understand the action of $\text{Aut}(V)$ after tensoring with $L$, which might or might not be sufficient, depending on the details of your problem.