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Let $M$ be a $R$-linear Chow motif over a field $k$ that is perfect but not necessarily algebraically closed. Can one prove that $M$ is not a direct summand of itself (that is, $M\not\cong M\bigoplus N$ for every non-zero Chow motif $N$ over $k$)?

Is this statement clear for any particular $(k,R)$ (let us assume that the exponential characteristic of $k$ is invertible in $R$)? Do there exist any counterexamples (for torsion coefficients, probably)?

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    $\begingroup$ Yes, I am aware of this argument. And I am mostly interested in unconditional results and torsion coefficients. $\endgroup$
    – Mikhail Bondarko
    Commented Sep 5, 2021 at 18:05
  • $\begingroup$ One positive result is Lemma 1 in arxiv.org/pdf/0806.0173.pdf: it basically says that a motive with vanishing Chow groups over all field extensions is zero. However it is not sufficient for the positive answer to your question as Chow groups are infinite dimensional in general. $\endgroup$
    – Evgeny Shinder
    Commented Sep 5, 2021 at 22:06
  • $\begingroup$ Yet something like this may help in the setting I am interested in; thank you, Evgeny! $\endgroup$
    – Mikhail Bondarko
    Commented Sep 6, 2021 at 6:02
  • $\begingroup$ Such "Krull-Schmidt" type questions have an interpretation in terms of $K_0$ of the category of Chow motives about which very little is known. Some nontrivial tricks how to work with $K_0$ of an additive category are explained in Section 2 of Efimov's paper on L-equivalence arxiv.org/pdf/1707.08997.pdf. $\endgroup$
    – Evgeny Shinder
    Commented Sep 6, 2021 at 9:02
  • $\begingroup$ Thank you! I actually started from a $K_0$-formulation (no classes should vanish in it), but I did not read Efimov's paper.:) $\endgroup$
    – Mikhail Bondarko
    Commented Sep 6, 2021 at 12:07

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