Motives with modulus

TL;DR

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

Contribution

It constructs the category $ extbf{MDM}_{ ext{gm}}^{ ext{eff}}$ using proper modulus pairs, broadening the scope of motives to non-homotopy invariant cases.

Findings

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

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

Motives are associated with pairs of proper varieties and effective divisors.

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 \emph{proper modulus pairs}, that is, pairs of a proper $k$-variety $X$ and an effective divisor $D$ on $X$ such that $X \setminus |D|$ is smooth. To a modulus pair $(X, D)$ we associate its motive $M(X, D) \in \mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$. In some cases the 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.