Vol. 7, No. 1, 2022

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Motives with modulus, III: The categories of motives

Bruno Kahn, Hiroyasu Miyazaki, Shuji Saito and Takao Yamazaki

Vol. 7 (2022), No. 1, 119–178
Abstract

We construct and study a triangulated category of motives with modulus MDM gm eff over a field k that extends Voevodsky’s category DM gm eff in such a way as to encompass nonhomotopy invariant phenomena. In a similar way as DM gm eff is constructed out of smooth k-varieties, MDM gm eff is constructed out of proper modulus pairs, introduced in Part I of this work. To such a modulus pair we associate its motive in MDM gm eff. In some cases, the  Hom group in MDM gm eff between the motives of two modulus pairs can be described in terms of Bloch’s higher Chow groups.

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Keywords
motives, modulus, reciprocity
Mathematical Subject Classification
Primary: 19E15
Secondary: 14F42, 19D45, 19F15
Milestones
Received: 22 February 2021
Revised: 23 July 2021
Accepted: 9 August 2021
Published: 20 June 2022
Authors
Bruno Kahn
IMJ-PRG
CNRS
4 place Jussieu
Case 247
75252 Paris Cedex 5
France
Hiroyasu Miyazaki
NTT Institute for Fundamental Mathematics
Tokyo
Japan
RIKEN iTHEMS
Wako, Saitama
Japan
Shuji Saito
Graduate School of Mathematical Sciences
University of Tokyo
Tokyo
Japan
Takao Yamazaki
Institute of Mathematics
Tohoku University
Aoba
Sendai
Japan