Connectivity and Purity for Logarithmic Motives

Abstract

The goal of this article 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 ${\operatorname {\mathbf {logDM}^{eff}}}(k)$ is proved to be compatible with Voevodsky’s t-structure; that is, we show that the comparison functor $R^{{\overline {\square }}}\omega ^*\colon {\operatorname {\mathbf {DM}^{eff}}}(k)\to {\operatorname {\mathbf {logDM}^{eff}}}(k)$ is t-exact. The heart of the homotopy t-structure on ${\operatorname {\mathbf {logDM}^{eff}}}(k)$ 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.

Motives and homotopy theory in logarithmic geometry

“…In [6], Binda-Merici proved the log analog of Morel's connectivity theorem, together with a purity result for -local complexes of sheaves with log transfers. Using this, and adapting an argument due to Ayoub and Morel, they construct a homotopy t -structure on log DM eff (k, Λ).…”

“…The heart of the said homotopy t -structure is the Grothendieck abelian category CI ltr dNis of strictly -invariant sheaves with log transfers. Under resolution of singularities, the purity theorem of [6] implies the composite…”

Erratum To: Connectivity and Purity for Logarithmic Motives

“…To show faithfulness, it is enough to show that for all (resp., ), the unit map is injective. By [1, Theorem 5.10], we have that for all , and hence (resp., ) is injective. Because F is -local, the map u (resp., ) factors through (resp., ), which concludes the proof.…”

“…If Conjecture 0.3 holds, then the counit map is a monomorphism for all . In particular, this would imply that the natural map is injective, so we could proceed as in [1, Proposition 7.3.] to prove Conjecture 0.2 in the case with transfers.…”

“…The proof of [1,Lemma 7.2] contains a gap: the equality ω h 0 (Λ ltr (η,triv)) = ω h 0 (ω * Λ tr (η)) is false. Indeed one can check that for X ∈ Sm(k) proper,…”