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Journal Articles Annals of K-Theory Year : 2022

Motives with modulus, III: The categories of motives

Abstract

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.

Dates and versions

hal-03319234 , version 1 (11-08-2021)

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Bruno Kahn, Hiroyasu Miyazaki, Shuji Saito, Takao Yamazaki. Motives with modulus, III: The categories of motives. Annals of K-Theory, 2022, 7 (1), pp.119-178. ⟨10.2140/akt.2022.7.119⟩. ⟨hal-03319234⟩
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