# The Homology of Connective Morava $E$-theory with coefficients in $\mathbb{F}_p$

## TL;DR

This paper computes the homology of the connective Morava $E$-theory spectrum $e_n$ with coefficients in $_p$ for heights up to 4, and establishes the multiplicativity of a spectral sequence in this context.

## Contribution

It introduces a proof that the K"unneth spectral sequence is multiplicative for $E_3$-algebras and applies this to compute homology of $e_n$ for $n \,\leq\, 4$.

## Key findings

- Homology $H_*(e_n;\mathbb{F}_p)$ computed for $n \leq 4$
- K"unneth spectral sequence shown to be multiplicative for certain $E_3$-algebras
- Application over $BP$ as an $E_4$-algebra

## Abstract

Let $e_n$ be the connective cover of the Morava $E$-theory spectrum $E_n$ of height $n$. In this paper we compute its homology $H_*(e_n;\mathbb{F}_p)$ for any prime $p$ and $n \leq 4$ up to possible multiplicative extensions. In order to accomplish this we show that the K\"unneth spectral sequence based on an $E_3$-algebra $R$ is multiplicative when the $R$-modules in question are commutative $S$-algebras. We then apply this result by working over $BP$ which is known to be an $E_4$-algebra.