Connectivity and Purity for logarithmic motives

TL;DR

This paper extends the homotopy t-structure and purity results from classical motivic complexes to logarithmic motives, establishing compatibility with Voevodsky's structures and introducing new categories of sheaves.

Contribution

It develops the homotopy t-structure for logarithmic motives, proves an analogue of Morel's connectivity theorem, and constructs a new category of reciprocity sheaves for log schemes.

Findings

Proves the t-structure on log motives is compatible with Voevodsky's t-structure.

Establishes a purity statement for log sheaves with transfers.

Constructs a new category of reciprocity sheaves for logarithmic schemes.

Abstract

The goal of this paper is to extend the work of Voevodsky and Morel on the homotopy $t$-structure on the category of motivic complexes to the context of motives for logarithmic schemes. To do so, we prove an analogue of Morel's connectivity theorem and show a purity statement for $(\mathbf{P}^1, \infty)$-local complexes of sheaves with log transfers. The homotopy $t$-structure on $\mathbf{logDM}^{\textrm{eff}}(k)$ is proved to be compatible with Voevodsky's $t$-structure i.e. we show that the comparison functor $R^{\overline{\square}}\omega^*\colon \mathbf{DM}^{\textrm{eff}}(k)\to \mathbf{logDM}^{\textrm{eff}}(k)$ is $t$-exact. The heart of the homotopy $t$-structure on $\mathbf{logDM}^{\textrm{eff}}(k)$ is the Grothendieck abelian category of strictly cube-invariant sheaves with log transfers: we use it to build a new version of the category of reciprocity sheaves in the style of Kahn--Saito--Yamazaki and R\"ulling.