Motivic Integral p-adic cohomologies

Informazioni relative al progetto

MIPAC

ID dell’accordo di sovvenzione: 101103309

DOI

10.3030/101103309

Progetto concluso il 30 Settembre 2024

Data della firma CE 18 Aprile 2023

Data di avvio 1 Settembre 2023

Data di completamento 31 Agosto 2025

Finanziato da

• Marie Skłodowska-Curie Actions (MSCA)

Costo totale

Nessun dato

Contributo UE

€ 172 750,08

Coordinato da

UNIVERSITA DEGLI STUDI DI MILANO
Italy

Periodic Reporting for period 1 - MIPAC (Motivic Integral p-adic cohomologies)

Periodo di rendicontazione: 2023-09-01 al 2025-08-31

Sintesi del contesto e degli obiettivi generali del progetto

Motivic homotopy theory is a highly successful theory built on the idea of introducing modern methods from homotopy theory into algebraic geometry, considering the affine line A1 as the interval object. While it has proved to be extremely successful in solving long standing conjectures in the cohomology of varieties, like the Milnor and Bloch--Kato conjecture, A1-homotopy invariance is structurally not compatible with integral p-adic cohomologies.
The goal of Motivic Integral p-adic Cohomologies (MIPAC) was to develop a theory of motivic p-adic cohomology theories in the context of motives of logarithmic schemes, and link it with the work done so far on tame cohomology.

Lavoro eseguito dall’inizio del progetto fino alla fine del periodo coperto dalla relazione e principali risultati finora ottenuti

The main achievements of this project were two: the first one is the comparison between tame and log étale motives. In this, we showed that the comparison theorem between Nisnevich motives and dNis-log motives of Binda-Park-Østvær can be promoted to a comparison functor between tame and log étale motives. This gives to a big family of tame motivic spectra coming from logSH_\loget, where computations are much easier. This allowed also to show that there exists a motivic spectrum HZ/p^m representing mod p^m-motivic cohomology for all m. The other significant contribution is the notion of saturated descent and its application to log motivic invariants, in particular to logTHH and logTC: this allowed to deduce comparison theorems between log theories from the comparison theorems for usual theories, and it allowed to promote them to motivic comparisons, deducing properties that were not known before (Gysin sequences, PBF…).

Progressi oltre lo stato dell’arte e potenziale impatto previsto (incluso l’impatto socioeconomico e le implicazioni sociali più ampie del progetto fino ad ora)

The results described above are certainly progress beyond the state of the art and open a new gateway to motivic homotopy theory.

Comparison between tame and log motives

Saturated descent for THH