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.
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Notes
While the adjective rational in the term rational point refers to the base, its meaning in the term rational motivic point is different—it refers to the coefficients, and to the effect they have on rationalizing spaces.
Drawing on the analogy between spectra and abelian groups, an object of a stable monoidal infinity category equipped with a unit morphism and a single binary operation (associative up to coherent higher homotopies) is often called an algebra. Although \({\text {ChM}}(Z,\mathbb {Q})\) is not itself stable, it is a full subcategory of the homotopy category of a stable monoidal infinity category.
Actually, philosophically, a comonoid in Chow motives should only be called a motivic Chow coalgebra if it comes from geometry in some sense. In the present work, we will only consider comonoids which come from geometry in an obvious sense.
When X is not proper, one must take the cofiber of the complement inside a compactification; see Proposition 3.6 for the precise statement under stringent assumptions on X.
We place an \(\Omega ^\infty \) between the two limits in order to emphasize the fact that while the inner limit may be taken inside the category of spectra, the \(\Delta \)-shaped diagram of the outer limit contains morphisms which are not morphisms of spectra.
Indeed, from this point of view, a morphism \( C^*X \rightarrow \mathbb {Q}(0) \) in \({\text {DA}}(Z,\mathbb {Q})\) might be called a “rational linear motivic point”.
A point which may cause confusion is that the pointed set \(H^1\big ( \pi _1^R(Z), \pi _1^\text {un}(X)^R \big )\) is a cohomology set in the sense of group cohomology (that is, when considered as a functor of \(\pi _1^R(Z)\)), but is covariant as a functor of X.
We alert the reader to the fact that in topos theory, our \(i^*\) is usually denoted \(i_*\).
Although the category of motivic dga’s of [8] admits a well-behaved model structure, the language of infinity categories provides an elegant setting and a wealth of ready-made tools for our computations with homotopy limits.
There are two natural model structures, and as noted by Lurie, here we are free to choose either one.
There is also a purely motivic explanation. Motivic cohomology is also the cohomology of certain complexes of Zariski sheaves; \(\mathbb {Q}(0)\) corresponds to the constant sheaf \(\mathbb {Q}\) concentrated in degree 0.
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Acknowledgements
We heartily thank Shane Kelly for helpful conversations throughout the development of this work. We would also like to acknowledge our intellectual debt to Marc Levine; in particular, our notion of motivic Chow coalgebra and the question of its relationship to motivic dga’s grew out of conversations with him. We thank Vladimir Hinich as well as the anonymous referee for helpful comments and suggestions.
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Communicated by Vladimir Hinich.
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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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Dan-Cohen, I., Schlank, T. Morphisms of Rational Motivic Homotopy Types. Appl Categor Struct 29, 311–347 (2021). https://doi.org/10.1007/s10485-020-09618-6
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DOI: https://doi.org/10.1007/s10485-020-09618-6