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Morphisms of Rational Motivic Homotopy Types
Applied Categorical Structures ( IF 0.6 ) Pub Date : 2020-11-19 , DOI: 10.1007/s10485-020-09618-6
Ishai Dan-Cohen , Tomer Schlank

We investigate several interrelated foundational questions pertaining to the study of motivic dga's of Dan-Cohen--Schlank [8] and Iwanari [13]. In particular, we note that morphisms of motivic dga's can reasonably be thought of as a nonabelian analog of motivic cohomology. Just as abelian motivic cohomology is a homotopy group of a spectrum coming from K-theory, the space of morphisms of motivic dga's is a certain limit of such spectra; we give an explicit formula for this limit --- a possible first step towards explicit computations or dimension bounds. We also consider commutative comonoids in Chow motives, which we call ``motivic Chow coalgebras''. We discuss the relationship between motivic Chow coalgebras and motivic dga's of smooth proper schemes. As a small first application of our results, we show that among schemes which are finite \'etale over a number field, morphisms of associated motivic dga's are no different than morphisms of schemes. This may be regarded as a small consequence of a plausible generalization of Kim's relative unipotent section conjecture, hence as an ounce of evidence for the latter.

中文翻译:

有理动机同伦类型的态射

我们调查了与 Dan-Cohen--Schlank [8] 和 Iwanari [13] 的动机 dga 研究有关的几个相互关联的基础问题。特别是,我们注意到动机 dga 的态射可以合理地被认为是动机上同调的非阿贝尔类比。正如阿贝尔动机上同调是来自 K 理论的一个谱的同伦群,动机 dga 的态射空间是这种谱的一个极限;我们为这个限制给出了一个明确的公式——可能是迈向明确计算或维度界限的第一步。我们还考虑了 Chow 动机中的交换 comonoids,我们称之为“motivic Chow 代数”。我们讨论了motivic Chow 余代数和motivic dga 之间的关系。作为我们结果的一个小的首次应用,我们表明,在数域上有限的方案中,相关动机 dga 的态射与方案的态射没有区别。这可能被视为 Kim 的相对单能截面猜想的似是而非的概括的一个小结果,因此作为后者的一点证据。
更新日期:2020-11-19
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