Motives with modulus

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.

Proposition 3.5.3 is false: we thank Joseph Ayoub for helping us find this mistake. Since it is a building block of our theory, we decided to withdraw the paper

References in corpus (3)

• Birational motives, II: Triangulated birational motives
• Additive Chow groups of schemes
• Suslin homology of relative curves with modulus

Cited by in corpus (11)

• Reciprocity sheaves
• Motives with modulus, III: The categories of motives
• Reciprocity sheaves and logarithmic motives
• Gysin triangles in the category of motifs with modulus
• A remark on the tensor product of SC-reciprocity sheaves
• Cube invariance of higher Chow groups with modulus
• Semi-purity for cycles with modulus
• On extension of the motivic cohomology beyond smooth schemes
• Suslin's moving lemma with modulus
• Suslin homology of relative curves with modulus
• Grauert-Riemenschneider Vanishing via modulus sheaves with transfers