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 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2020
- DOI:
- 10.48550/arXiv.2012.08361
- arXiv:
- arXiv:2012.08361
- Bibcode:
- 2020arXiv201208361B
- Keywords:
-
- Mathematics - Algebraic Geometry;
- Mathematics - K-Theory and Homology;
- 14F08 (Primary);
- 18E35;
- 14F42;
- 18G80 (Secondary)
- E-Print:
- A gap was found in a proof on the last section. We modified the statement to a weaker form