On the Coefficients of $(\mathbb{Z}/p{)}^n$-Equivariant Ordinary Cohomology with Coefficients in $\mathbb{Z}/p$

Abstract

This note contains a generalization to $p>2$ of the authors' previous calculations of the coefficients of $(\mathbb{Z}/2{)}^n$-equivariant ordinary cohomology with coefficients in the constant $\mathbb{Z}/2$-Mackey functor. The algberaic results by S.Kriz allow us to calculate the coefficients of the geometric fixed point spectrum ${\Phi }^{(\mathbb{Z}/p{)}^n}H\mathbb{Z}/p$, and more generally, the $\mathbb{Z}$-graded coefficients of the localization of $H\mathbb{Z}/p_{(\mathbb{Z}/p{)}^n}$ by inverting any chosen set of embeddings $S^0\to S^{{\alpha }_i}$ where ${\alpha }_i$ are non-trivial irreducible representations. We also calculate the $RO(G{)}^{+}$-graded coefficients of $H\mathbb{Z}/p_{(\mathbb{Z}/p{)}^n}$, which means the cohomology of a point indexed by an actual (not virtual) representation. (This is the "non-derived" part, which has a nice algebraic description.)