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 $kq$.
Categories: math.AT