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Let $k$ be an algebraically closed field. Let $S$ be a homotopy invariant $\mathbb{Q}$-linear sheaf with transfers in the sense of Voevodsky–Suslin, and assume that the dimension of $S(U)$ (over $\mathbb{Q}$) is bounded for a constant $c$ if $U$ is a smooth (connected) $k$-variety. Is it known that $S$ is constant?

I probably have a somewhat clumsy proof of this fact using Suslin rigidity-type arguments; yet I wonder which facts related to this one are already known. Moreover, I am actually interested in the extension of $S$ to pro-smooth (say, affine) $k$-schemes; and my finite dimensionality assumption corresponds to the finite dimensionality of $S(\operatorname{Spec} K)$, where $K$ is an algebraically closed field extension of $k$ of infinite transcendence degree. Consequently, I would also like to know which statements should I cite to deal with "colimit extensions" of this sort; is this section 8.13 of EGA 4?

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I have proved this statement as Lemma 5.1.3 at Bondarko and Sosnilo - On Chow-weight homology of geometric motives. Comments are very welcome!

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  • $\begingroup$ Why not on the arXiv, or at least your web-page? (I always find the maybe-available-maybe-not interface of ResearchGate confusing and slightly scammy seeming.) $\endgroup$
    – LSpice
    Commented Jul 1, 2020 at 15:25
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    $\begingroup$ I should create my webpage, but I didn't find time to do this yet.(: As about arxiv: there is an alternative version of our paper there. Neither of the versions is currently "strictly better" than another one; thus I would prefer to keep them at distinct places. Maybe, I will merge the versions eventually. $\endgroup$
    – Mikhail Bondarko
    Commented Jul 1, 2020 at 15:35
  • $\begingroup$ I noticed on the arXiv that there is a paper with different authors: Bondarko and Zumallagov - On Chow-weight homology of motivic complexes and its relation to motivic homology. Is that the one to which you refer? $\endgroup$
    – LSpice
    Commented Jul 1, 2020 at 15:37
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    $\begingroup$ No; this is the next paper on the subject; it is very much different from both versions of the text that I refer above (the other one is arxiv.org/abs/1411.6354). My new co-author is Kumallagov.:) $\endgroup$
    – Mikhail Bondarko
    Commented Jul 1, 2020 at 17:17
  • $\begingroup$ Oops, sorry for misspelling Kumallagov's name! $\endgroup$
    – LSpice
    Commented Jul 1, 2020 at 17:32

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