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CONNECTIVITY AND PURITY FOR LOGARITHMIC MOTIVES

Part of: (Co)homology theory

Published online by Cambridge University Press:  14 June 2021

Federico Binda Open the ORCID record for Federico Binda [Opens in a new window]  and
Alberto Merici
Federico Binda
Affiliation:
Dipartimento di Matematica “Federigo Enriques”, Università degli Studi di Milano Via Cesare Saldini 50, Milano 20133, Italy (federico.binda@unimi.it)
Alberto Merici
Affiliation:
Institut für Mathematik, Universität Zurich, Winterthurerstr. 190, CH-8057 Zürich, Switzerland (alberto.merici@math.uzh.ch)
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Abstract

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The goal of this article is to extend the work of Voevodsky and Morel on the homotopy t-structure on the category of motivic complexes to the context of motives for logarithmic schemes. To do so, we prove an analogue of Morel’s connectivity theorem and show a purity statement for $({\mathbf {P}}^1, \infty )$ -local complexes of sheaves with log transfers. The homotopy t-structure on ${\operatorname {\mathbf {logDM}^{eff}}}(k)$ is proved to be compatible with Voevodsky’s t-structure; that is, we show that the comparison functor $R^{{\overline {\square }}}\omega ^*\colon {\operatorname {\mathbf {DM}^{eff}}}(k)\to {\operatorname {\mathbf {logDM}^{eff}}}(k)$ is t-exact. The heart of the homotopy t-structure on ${\operatorname {\mathbf {logDM}^{eff}}}(k)$ is the Grothendieck abelian category of strictly cube-invariant sheaves with log transfers: we use it to build a new version of the category of reciprocity sheaves in the style of Kahn-Saito-Yamazaki and Rülling.

Keywords

MotivesLogarithmic SchemesCohomology Theories

MSC classification

Secondary: 14F42: Motivic cohomology; motivic homotopy theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 22 , Issue 1 , January 2023 , pp. 335 - 381
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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