Motives and homotopy theory in logarithmic geometry

Abstract

This document is a short user's guide to the theory of motives and homotopy theory in the setting of logarithmic geometry. We review some of the basic ideas and results in relation to other works on motives with modulus, motivic homotopy theory, and reciprocity sheaves.Résumé. Ce document est un petit guide d'utilisation de la théorie des motifs et de la théorie de l'homotopie dans le cadre de la géométrie logarithmique. Nous passons en revue certaines des idées de base et des résultats en relation avec d'autres travaux sur les motifs avec module, théorie de l'homotopie motivique, et les faisceaux de réciprocité.

On the cohomology of reciprocity sheaves

“…In the modulus setting, where smooth schemes get replaced by ‘compactifications’ , we need then to ‘compactify’ without changing its ‘homotopy type’, and the pair does the job. For reduced modulus, the formula in Theorem 7.16 is also witnessed in the logarithmic setting (see [BPØ22, Chapter 7, Section 5]).…”

“…In [BPØ22], Park, Østvær and the first author recently introduced a triangulated category of logarithmic motives over a field k . Similar in spirit to Voevodsky’s construction, the starting point is the category of log smooth (fs)-log schemes over k , promoted then to a category of correspondences.…”

“…A theorem of Saito (see [Sai20b]) shows that there exists a fully faithful exact functor: such that is strictly -invariant in the sense of [BPØ22, Definition 5.2.2], where the target is the category of dividing Nisnevich sheaves with log transfers on (see [BPØ22, Section 2.4]). This shows that Nisnevich cohomology of reciprocity sheaves is representable in .…”

Connectivity and Purity for Logarithmic Motives

“…Voevodsky's category of motives over a field has been recently extended to the setting of logarithmic algebraic geometry in [7]. The basic objects in this context are no longer smooth k-schemes but rather fine and saturated log schemes, log smooth over a base considered with trivial log structure (typically, the base is a perfect field).…”

“…The category of log motives logDM eff (k,Λ) (with transfers) is then defined as the homotopy category of the (dNis, )-local model structure on the category of (unbounded) chain complexes of presheaves with logarithmic transfers, C(PSh ltr (k,Λ)), for Λ a ring of coefficients. See [7,[4][5] and Section 2 for more details. The variant without transfers will be denoted logDA eff (k,Λ), and it is obtained as Bousfield localisation of the category of (unbounded) chain complexes of presheaves C(PSh log (k,Λ)).…”