## Motives with modulus, III: the categories of motives

#### Bruno Kahn, Hiroyasu Miyazaki, Shuji Saito and Takao Yamazaki

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

Reference :

ArXiv link : https://arxiv.org/abs/2011.11859

Hal link : https://hal.archives-ouvertes.fr/hal-03319234v1

Comments on this version : To appear in the Ann. of K-theory

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Last version : 11-08-2021