$\mathbb{C}$-Motivic Eilenberg-Mac Lane Spectra as $\mathrm{MGL}$-Motivic Thom Spectra and Normed $\mathbb{G}_m$-Hochschild Homology

Source abstract

We exhibit the $p$-complete $\mathbb{C}$-motivic Eilenberg-Mac Lane spectra $\mathrm{M}\mathbb{F}_p$ and $(\mathrm{M}\mathbb{Z})_p^{\wedge}$ as $\mathbb{E_3}$-$\mathrm{MGL}_p^{\wedge}$-motivic Thom spectra over the motivic $\mathbb{G}_m$-loop spaces $Ω_{\mathbb{G}_m}\mathrm{BGL}$ and $Ω_{\mathbb{G}_m}\mathrm{BSL}$, respectively. Furthermore, for $p=2$, we compute the $\mathrm{MGL}_2^{\wedge}$-relative $\mathbb{G}_m$-Hochschild Homology $\mathbb{G}_m \odot^{\mathrm{MGL}_2^{\wedge}} {-}$, recently introduced by Bachmann-Burklund-Dundas-Østvær, of $\mathrm{M}\mathbb{F}_2$ and $(\mathrm{M}\mathbb{Z})_2^{\wedge}$ to be $\mathrm{M}\mathbb{F}_2[\mathrm{BGL}]$ and $(\mathrm{M}\mathbb{Z})_2^{\wedge}[\mathrm{BSL}]$ as $\mathbb{E}_2$-$\mathrm{M}\mathbb{F}_2$-algebras and $\mathbb{E}_2$-$(\mathrm{M}\mathbb{Z})_2^{\wedge}$-algebras, respectively.