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 $\mathbf{logDM}^{\textrm{eff}}(k)$ 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 cubeinvariant sheaves with log transfers: we use it to build a new version of the category of reciprocity sheaves in the style of KahnSaitoYamazaki and Rülling.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 DOI:
 10.48550/arXiv.2012.08361
 arXiv:
 arXiv:2012.08361
 Bibcode:
 2020arXiv201208361B
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  KTheory and Homology;
 14F08 (Primary);
 18E35;
 14F42;
 18G80 (Secondary)
 EPrint:
 A gap was found in a proof on the last section. We modified the statement to a weaker form