The Motivic Cofiber of $\tau$

Abstract

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 ${\pi }_{\ast ,\ast }C\tau \cong {\mathrm{Ext}}_{B{P}_{\ast }BP}^{\ast ,\ast }(B{P}_{\ast },B{P}_{\ast })$ 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 $H{\mathbb{F}}_2$, the mod 2 Moore spectrum $S^{0,0}/2$ and the connective hermitian $K$-theory spectrum $kq$.