$τ$的动机共纤
The Motivic Cofiber of $τ$
摘要：
考虑运动球的模2上同调中的Tate扭曲$τ\in H^{0,1}(S^{0,0})$。在2-完备化之后，动机Adams谱序列将该元素实现为具有共纤$Cτ$的映射$τ\colon S^{0,-1} \to S^{0,0}$。我们表明，这种motivic 2-细胞复合物可以被赋予一个独特的$E_{\infty}$环结构。此外，这将已知的同构$π_{\ast,\ast} Cτ\cong \mathrm{Ext}^{\ast,\ast}_{BP_{\ast}BP}(BP_{\ast},BP_{\ast})$提升为环的同构，其也保持更高的积。然后我们考虑封闭的对称monoidal范畴$({ }_{Cτ}\textbf{Mod}, - \wedge_{Cτ} -)$的Betti实现的核心生活。给定一个动机谱$X$，$Cτ$诱导的谱$X \wedge Cτ$通常比$X$本身表现得更好，更容易理解.我们在模2 Eilenberg-Maclane谱$H\mathbb{F}_2$、模2 Moore谱$S^{0,0}/2$和连接厄米特$K$ -理论谱$kq$的例子中具体说明了这个概念。
英文摘要：
Consider the Tate twist $τ\in H^{0,1}(S^{0,0})$ in the mod 2 cohomology of the motivic sphere. After 2-completion, the motivic Adams spectral sequence realizes this element as a map $τ\colon S^{0,-1} \to S^{0,0}$, with cofiber $Cτ$. We show that this motivic 2-cell complex can be endowed with a unique $E_{\infty}$ ring structure. Moreover, this promotes the known isomorphism $π_{\ast,\ast} Cτ\cong \mathrm{Ext}^{\ast,\ast}_{BP_{\ast}BP}(BP_{\ast},BP_{\ast})$ to an isomorphism of rings which also preserves higher products. We then consider the closed symmetric monoidal category $({ }_{Cτ}\textbf{Mod}, - \wedge_{Cτ} -)$ which lives in the kernel of Betti realization. Given a motivic spectrum $X$, the $Cτ$-induced spectrum $X \wedge Cτ$ 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 $kq$.
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