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Milnor-Witt sheaves and modules

Abstract : We generalize Rost's theory of cycle modules using the Milnor-Witt K-theory instead of the classical Milnor K-theory. We obtain a (quadratic) setting to study general cycle complexes and their (co)homology groups.Moreover, we prove that the heart of Morel-Voevodsky stable homotopy category over a perfect field (equipped with its homotopy t-structure) is equivalent to the category of Milnor-Witt cycle modules.Finally, we explore a conjecture of Morel about the Bass-Tate transfers defined on the contraction of a homotopy sheaf and prove that the conjecture is true with rational coefficients. We also study the relations between (contracted) homotopy sheaves, sheaves with generalized transfers and MW-homotopy sheaves, and prove an equivalence of categories. As applications, we describe the essential image of the canonical functor that forgets MW-transfers and use theses results to discuss the conservativity conjecture in A1-homotopy due to Bachmann and Yakerson.
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https://tel.archives-ouvertes.fr/tel-03225375
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Submitted on : Wednesday, May 12, 2021 - 2:34:14 PM
Last modification on : Friday, March 25, 2022 - 9:40:32 AM
Long-term archiving on: : Friday, August 13, 2021 - 6:37:39 PM

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Niels Feld. Milnor-Witt sheaves and modules. Mathematics [math]. Université Grenoble Alpes [2020-..], 2021. English. ⟨NNT : 2021GRALM001⟩. ⟨tel-03225375⟩

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