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: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 of C\tau -modules ( _{C\tau} 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\Bbb F_2 , the mod 2 Moore spectrum S^{0,0}/2 and the connective hermitian K -theory spectrum kq .