The Motivic Cofiber of $\tau$

Consider the Tate twist $\tau in H^{0,1}({S^{0,0})$ in the mod 2 cohomology of the motivic sphere. After 2-completion, the motivic Adams spectral sequence realizes this element as a map $\tau \colon S^{0,-1} \lto {S^{0,0}$, with cofiber $C \tau$. We show that this motivic 2-cell complex can be endowed with a unique $E_{\infty}$ ring structure. Moreover, this promotes the known isomorphism ${\pi_{\ast,\ast} C \tau \cong \Ext^{\ast,\ast}_{BP_{\ast}BP}(BP_{\ast},BP_{\ast})$ to an isomorphism of rings which also preserves higher products....
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