The homology of connective Morava $E$-theory with coefficients in $\mathbb{F}_p$

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 $H_\ast (e_n; \mathbb{F}_p)$ for any prime $p$ and $n \leqslant 4$ up to possible multiplicative extensions. In order to accomplish this we show that the Künneth spectral sequence based on an $\mathbb{E}_3$-algebra $R$ is multiplicative when the $R$-modules in question are commutative Salgebras. We then apply this result by working over $BP$ which is known to be an $\mathbb{E}_4$-algebra.

Keywords

highly structured ring spectra, Künneth spectral sequence, Morava $E$-theory

2010 Mathematics Subject Classification

Primary 55P43, 55T15. Secondary 55N20.

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Both authors were supported by the German Research Council DFG-GRK 1916. Moreover, the first author was supported by the German Research Council, grant KA 4128/2-1.

Received 9 July 2019

Received revised 25 February 2021

Accepted 1 June 2021

Published 11 October 2023