Abstract
AbstractThe goal of this article is to extend the work of Voevodsky and Morel on the homotopyt-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 homotopyt-structure on${\operatorname {\mathbf {logDM}^{eff}}}(k)$is proved to be compatible with Voevodsky’st-structure; that is, we show that the comparison functor$R^{{\overline {\square }}}\omega ^*\colon {\operatorname {\mathbf {DM}^{eff}}}(k)\to {\operatorname {\mathbf {logDM}^{eff}}}(k)$ist-exact. The heart of the homotopyt-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.
Highlights
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
We prove an analogue of Morel’s connectivity theorem and show a purity statement for (P1,∞)-local complexes of sheaves with log transfers
The heart of the homotopy t-structure on logDMeff (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 Rulling
Summary
Voevodsky’s category of motivic complexes over a perfect field k is based on a simple idea: most cohomology theories for smooth k-schemes are insensitive to the affine line; that is, they satisfy A1-homotopy invariance. The good properties of the category of strictly -invariant sheaves CIldtNr is, deduced from the identification with the heart of the homotopy t-structure, allow us to make a further comparison with the category RSCNis of reciprocity sheaves of Kahn-SaitoYamazaki This is an abelian subcategory of the category of Nisnevich sheaves with transfers ShvtNris(k), whose objects satisfy a certain restriction on their sections inspired by the Rosenlicht-Serre theorem on reciprocity for morphisms from curves to commutative algebraic groups [30, III]. If we denote by iRSC the inclusion RSCNis ⊂ ShvtNris, we can identify the functor Log of (1.4.1) with the composite ωlCogI ◦ iRSC, where ωlCogI is the right adjoint to ωCloIg. The category LogRec seems to share many of the properties of RSCNis: in the rest of Section 7 we discuss some of them, in particular in relationship with the monoidal structure.
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