Motives with modulus, III: The categories of motives

TL;DR

This paper introduces a new triangulated category of motives with modulus that extends Voevodsky's category to include non-homotopy invariant phenomena, constructed from proper modulus pairs.

Contribution

It constructs the category $ extbf{MDM}_{ ext{gm}}^{ ext{eff}}$ of motives with modulus, extending existing frameworks to encompass broader algebraic phenomena.

Findings

Hom groups can be described using Bloch's higher Chow groups in some cases

The category extends Voevodsky's motives to non-homotopy invariant cases

Motives are associated to proper modulus pairs

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.