Connectivity and Purity for logarithmic motives

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 ${\mathbf{logDM}}^{\text{eff}}(k)$ is proved to be compatible with Voevodsky's $t$-structure i.e. we show that the comparison functor $R^{\overline{\square }}{\omega }^{\ast }:{\mathbf{DM}}^{\text{eff}}(k)\to {\mathbf{logDM}}^{\text{eff}}(k)$ is $t$-exact. The heart of the homotopy $t$-structure on ${\mathbf{logDM}}^{\text{eff}}(k)$ 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 Rülling.

A gap was found in a proof on the last section. We modified the statement to a weaker form

References in corpus (1)

• Cancellation theorems for reciprocity sheaves