The Chow $t$-structure on the $\infty$-category of motivic spectra
The Chow $t$-structure on the $\infty$-category of motivic spectra
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DOI：
10.4007/annals.2022.195.2.5
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发表时间：
2020-12
影响因子：
4.9
通讯作者：
Tom Bachmann;Hana Jia Kong;Guozhen Wang;Zhouli Xu
中科院分区：
文献类型：
--
作者：
Tom Bachmann;Hana Jia Kong;Guozhen Wang;Zhouli Xu
我们在任意基域$k$上的动机谱$SH(K)$范畴上定义了Chow$t$-结构。当$k$的指数特征倒置时，我们确定了这个$t$结构$SH(K)^{c\HeartSuit}$的核心。限制到胞胞子范畴，我们将Chow心$SH(K)^{cell，c\HeartSuit}$定义为偶分次$MU_(2*)MU$-余模范畴。进一步，我们证明了Chow截断球谱上的模的$\inty$-范畴是代数的。我们的结果在三个方面推广了Gheorghe-Wang-Xu中的结果：积分结果；除了$C$之外的所有基场；整个$\inty$-动机谱范畴$SH(K)$，而不是只包含某些细胞对象的子范畴。我们还讨论了利用与Chow$t$-结构相关的Postnikov塔和$k$上的Motivic Adams谱序列计算任意基域$k$上的(p-完全)球面的Motivic稳定同伦群的策略。
We define the Chow $t$-structure on the $\infty$-category of motivic spectra $SH(k)$ over an arbitrary base field $k$. We identify the heart of this $t$-structure $SH(k)^{c\heartsuit}$ when the exponential characteristic of $k$ is inverted. Restricting to the cellular subcategory, we identify the Chow heart $SH(k)^{cell, c\heartsuit}$ as the category of even graded $MU_{2*}MU$-comodules. Furthermore, we show that the $\infty$-category of modules over the Chow truncated sphere spectrum is algebraic. Our results generalize the ones in Gheorghe--Wang--Xu in three aspects: To integral results; To all base fields other than just $C$; To the entire $\infty$-category of motivic spectra $SH(k)$, rather than a subcategory containing only certain cellular objects. We also discuss a strategy for computing motivic stable homotopy groups of (p-completed) spheres over an arbitrary base field $k$ using the Postnikov tower associated to the Chow $t$-structure and the motivic Adams spectral sequences over $k$.