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LSpice
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I have proved this statement as Lemma 5.1.3 at https://www.researchgate.net/publication/340849991_On_Chow-weight_homology_of_geometric_motives Comments are very welcome!

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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Mikhail Bondarko
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I have proved this statement as Lemma 5.1.3 at https://www.researchgate.net/publication/340849991_On_Chow-weight_homology_of_geometric_motives Comments are very welcome!