Morphisms of Rational Motivic Homotopy Types
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Morphisms of Rational Motivic Homotopy Types Ishai Dan-Cohen1
· Tomer Schlank2
Received: 26 November 2019 / Accepted: 3 November 2020 © Springer Nature B.V. 2020
Abstract We investigate several interrelated foundational questions pertaining to the study of motivic dga’s of Dan-Cohen and Schlank (Rational motivic path spaces and Kim’s relative unipotent section conjecture. arXiv:1703.10776) and Iwanari (Motivic rational homotopy type. arXiv:1707.04070). 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 étale 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. Keywords Rational homotopy · Motives · Algebraic K-theory · Infinity categories Mathematics Subject Classification 14C15 · 55P62
Communicated by Vladimir Hinich. I.D. was supported by an ISF Grant (No. 87590031) for work “Around Kim’s conjecture: from homotopical foundations to algorithmic applications.” T.S. was supported by an Alon Fellowship.
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Ishai Dan-Cohen [email protected] Tomer Schlank [email protected]
1
Department of Mathematics, Ben-Gurion University of the Negev, Beersheba, Israel
2
Einstein Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel
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I. Dan-Cohen, T. Schlank
1 Introduction 1.1 Overview Let X be a smooth scheme over a base Z . When k is a number field and Z = Spec k is the associated “number scheme”, a section of the projection X→Z is often referred to as a rational point. When instead Z is an open subscheme of Spec Ok (an “open integer scheme”), such a section is often called an integral point. Regardless of the particular setting, our theory of motivic dga’s gives rise to a weaker notion of point which we call rational motivic point. This theory is based on a certain functor ∗ C ∗ = CDMdga : Sm(Z )op → DMdga(Z , Q)
from the category of smooth schemes over Z to a category of motivic dga’s, which may be thought of as a motivic avatar of rational homotopy theory. To construct it, we consider a model M(Z , Q) for the tensor triangulated category DA(Z , Q) of motivic complexes over Z with Q coefficients. The category Mdga(Z , Q) of commutative monoids in M(Z , Q) inherits a model structure, and we let DMdga(Z , Q) be its homotopy cat
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