The $\mathbb{Z}/p$-equivariant spectrum $BP\mathbb{R}$ for an odd prime $p$

TL;DR

This paper constructs a $Z/p$-equivariant version of the $BPR$ spectrum for odd primes, extending previous work and confirming some conjectured properties, using an equivariant Brown-Peterson tower approach.

Contribution

It introduces a new $Z/p$-equivariant $BPR$ spectrum for odd primes, based on an equivariant Brown-Peterson tower and a $Z/p$-equivariant Steenrod algebra, with various variants and comparisons.

Findings

Spectrum has properties conjectured by Hill, Hopkins, and Ravenel.

Provides a $Z/p$-equivariant analogue of $BPR$ for odd primes.

Describes variants and comparisons with other spectra.

Abstract

In the present paper, we construct a $\mathbb{Z}/p$-equivariant analog of the $\mathbb{Z}/2$-equivariant spectrum $BP\mathbb{R}$ previously constructed by Hu and Kriz. We prove that this spectrum has some of the properties conjectured by Hill, Hopkins, and Ravenel. Our main construction method is an $\mathbb{Z}/p$-equivariant analog of the Brown-Peterson tower of $BP$, based on a previous description of the $\mathbb{Z}/p$-equivariant Steenrod algebra with constant coefficients by the authors. We also describe several variants of our construction and comparisons with other known equivariant spectra.