Motivic Integral p-adic cohomologies

Project Information

MIPAC

Grant agreement ID: 101103309

DOI

10.3030/101103309

Project terminated on 30 September 2024

EC signature date 18 April 2023

Start date 1 September 2023

End date 31 August 2025

Funded under

• Marie Skłodowska-Curie Actions (MSCA)

Total cost

No data

EU contribution

€ 172 750,08

Coordinated by

UNIVERSITA DEGLI STUDI DI MILANO
Italy

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

Reporting period: 2023-09-01 to 2025-08-31

Summary of the context and overall objectives of the project

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

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…).

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

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