Consider the connective $K$-theory spectrum $ku$. Let $H\mathbb{Z}$ be the Eilenberg-MacLane spectrum and $H\mathbb{F}_p$ be the mod-$p$ Eilenberg-MacLane spectrum.

Is it known what $ku^{*}(H\mathbb{Z})$, or $ku^{*}(H\mathbb{F}_{p})$, is?

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    $\begingroup$ I think "motivic" should be in the title, because that's an essential part of the question! $\endgroup$ – David White Mar 8 at 13:54
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    $\begingroup$ If you let me replace these spectra with their $p$-completions, then we have that $ku_p^*(H\mathbb{F}_p) \approx H\mathbb{F}_p^{*-4}(ku_p)$ and $ku_p^*(H\mathbb{Z})\approx H\mathbb{Z}_p^{*-3}(ku_p)$. $\endgroup$ – Charles Rezk Mar 8 at 18:37
  • $\begingroup$ Dear @CharlesRezk. Do you have any reference for this? I would really appreciate it. $\endgroup$ – user438991 Mar 8 at 21:03
  • $\begingroup$ @DavidWhite Why? The question seems to ask specifically "normal" connective K theory of normal Eilenberg Maclane spectra? $\endgroup$ – user43326 Mar 12 at 7:51
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    $\begingroup$ The OP edited later to remove motivic from the body $\endgroup$ – David White Mar 12 at 10:51

Charles Rezk already answered this in the comments; I'll just expand on what he wrote. This paper discusses what's now known as Mahowald-Rezk duality; this is a version of Anderson duality that takes into account "additive degeneration". I'll briefly recall some facts from that paper before mentioning how to use it to prove Rezk's claim. As in their paper, I'll be $p$-completing everything in sight for some fixed prime $p$.

Let $A^\ast$ (resp. $A_\ast$) denote the mod $p$ (dual) Steenrod algebra. Define a functor $\widetilde{J}$ from left $A_\ast$-comodules to left $A^\ast$-modules via $\widetilde{J}(M) = \mathrm{Hom}_{A_\ast}(A_\ast, M)$, and if $M$ is finitely generated, let $\widetilde{I}(M)$ denote the dual $\widetilde{J}(M)^\vee$. In section 8 of their paper, Mahowald and Rezk show that if $X$ is a fp-spectrum of fp-type $\leq n$ (intuitively, this is the statement that $X$ is in the thick subcategory generated by $\mathrm{BP}\langle n\rangle$; I don't know if being an fp-spectrum of fp-type $\leq n$ is equivalent to this), then there is a spectrum $W_n X$ such that $\mathrm{H}_\ast(W_n X) = \widetilde{I}(\mathrm{H}_\ast X)$. The spectrum $ku$ (again, $p$-completed) is a fp-spectrum of fp-type $\leq 1$, so there is a spectrum $W_1 ku$ such that

$$\mathrm{H}_\ast(W_1 ku) = \widetilde{I}(\mathrm{H}_\ast ku) = \mathrm{Hom}_{A_\ast}(A_\ast, \mathrm{H}_\ast ku)^\vee = [\mathrm{H}\mathbf{F}_p, ku]_\ast^\vee,$$

where the final identification comes from Lemma 6.2 of their paper. It therefore remains to compute $W_1 ku$, which is done in Corollary 9.3 of their paper: $W_1 ku = \Sigma^4 ku$. It follows that $\mathrm{H}^{\ast-4}(ku;\mathbf{F}_p) = ku^\ast(\mathrm{H}\mathbf{F}_p)$. The claim about the $ku$-cohomology of $\mathrm{H}\mathbf{Z}$ can be deduced from this using the cofiber sequence $\mathrm{H}\mathbf{Z} \xrightarrow{p} \mathrm{H}\mathbf{Z} \to \mathrm{H}\mathbf{F}_p$.


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