Solid Motives · Proétale and solid motivic sheaves

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Project objective

The main goal of this project is to construct an enlargement of Voevodsky's category of étale motives in which topological rings can be considered as coefficients. In this formalism, ℓ-adic realization functors will be particularly well-behaved and simply given by changing the ring of coefficients to the topological ring of ℓ-adic numbers. This will be achieved by merging Voevodsky's ideas with the recent work of Clausen–Scholze on condensed mathematics. A key technical challenge is to generalize the notion of a solid abelian group to a much broader context so that it also applies to categories of motivic nature. One of the key ingredients I will use is the machinery of internal higher categories, developed in joint work with Martini.I will prove that the categories I construct possess many of the convenient properties one expects. Specifically, I will show that they are equipped with a well-behaved formalism of six operations and that a suitable version of the Suslin–Voevodsky rigidity theorem holds. I then intend to use this new machinery to address the question of the independence of ℓ in Betti numbers for varieties of positive characteristic—a long-standing and difficult open problem—following ideas of Cisinski. The key property I will exploit is that, in this new formalism, the motive representing ℓ-adic cohomology is stable under base change, which is not the case in the setting of usual étale motives.I will also use some of the convenient properties of ℓ-adic realization in this setup to construct new categories of Nori motives with spectral coefficients, providing new evidence for a conjecture of Ayoub.

Original text from CORDIS.

Participants

• KOBENHAVNS UNIVERSITET · KOBENHAVNCoordinatorDenmark

Links

• View on CORDIS
• DOI: 10.3030/101276431