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Borel–Moore motivic homology and weight structure on mixed motives

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Abstract

By defining and studying functorial properties of the Borel–Moore motivic homology, we identify the heart of Bondarko–Hébert’s weight structure on Beilinson motives with Corti–Hanamura’s category of Chow motives over a base, therefore answering a question of Bondarko.

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Notes

  1. We will recall the definition of motivic cohomology groups in Definition 4.1.

  2. Recall that this means that the canonical closed immersion from the normal cone of \(Z'\) in \(X'\) to the pull-back by p of the normal cone of Z in X [20, B.6.1] is an isomorphism.

  3. \(\phi _X\) is an element of \(H^{BM}_{1,0}({\mathbb {G}}_{m,X}/X) \simeq {\mathbb {Z}}\times {\mathcal {O}}^*(X)\), which corresponds to the fundamental parameter of \({\mathbb {G}}_m\), see Lemma 3.6.

  4. The index i here means the i-th graded part of the total term.

  5. In view of the theory of generic motives, \(\hat{H}^{BM}_{n,i}(x)\) is nothing else but the Borel–Moore homology of the generic motive associated to the residue field of x, which only depends on the residue field, see [13].

  6. The isomorphism is constructed in the following way: there is a natural map \(\hat{H}^{BM}_{p,q}(z')\rightarrow \hat{H}^{BM}_{p,q}(z)\) defined in a similar way as the map \(\phi ^{*}_t\) above, and it is an isomorphism because f induces an isomorphism between the residue fields of z and \(z'\), therefore induces an isomorphism between the associated pro-schemes.

  7. Hébert showed that we can replace projective S-schemes by all proper S-schemes [21, Lemme 3.1].

  8. More generally, this assertion holds if S is separated of finite type over an excellent noetherian separated base scheme of dimension at most two (such a scheme is said to be “reasonable” in [7, Definition 1.1.1]).

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Acknowledgments

I would like to express my gratitude to my thesis advisor, Frédéric Déglise, who kindly shared the basic ideas of this article and helped me greatly during its development. I would also like to thank Mikhail Bondarko for very helpful discussion and advice on a preprint version.

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    Jin Fangzhou

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Fangzhou, J. Borel–Moore motivic homology and weight structure on mixed motives. Math. Z. 283, 1149–1183 (2016). https://doi.org/10.1007/s00209-016-1636-7

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