Abstract

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

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