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 $\tau \colon S^{0,-1} \to S^{0,0}$, 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 $\pi_{\ast,\ast} C\tau \cong \mathrm{Ext}^{\ast,\ast}_{BP_{\ast}BP}(BP_{\ast},BP_{\ast})$ to an isomorphism of rings which also preserves higher products. We then consider the closed symmetric monoidal category $({ }_{C\tau}\textbf{Mod}, - \wedge_{C\tau} -)$ which lives in the kernel of Betti realization. Given a motivic spectrum $X$, the $C\tau$-induced spectrum $X \wedge C\tau$ is usually better behaved and easier to understand than $X$ itself. We specifically illustrate this concept in the examples of the mod 2 Eilenberg-Maclane spectrum $H\mathbb{F}_2$, the mod 2 Moore spectrum $S^{0,0}/2$ and the connective hermitian $K$-theory spectrum $kq$.