Morphisms of rational motivic homotopy types

Ishai Dan-Cohen; Tomer Schlank

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arXiv · 1811.06365

Morphisms of rational motivic homotopy types

Ishai Dan-Cohen·Tomer Schlank

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Abstract

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.

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Ishai Dan-Cohen, Tomer Schlank. 2018-11-15. Morphisms of rational motivic homotopy types. https://arxiv.org/abs/1811.06365

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