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.

Minor corrections. To appear in Ann. of K-theory

References in corpus (3)

• Reciprocity sheaves and logarithmic motives
• A motivic homotopy theory without ${\mathbb{A}}^1$ invariance
• Triangulated categories of logarithmic motives over a field