Periodic Reporting for period 1 - MIPAC (Motivic Integral p-adic cohomologies)
Reporting period: 2023-09-01 to 2025-08-31
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.
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.
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…).
The results described above are certainly progress beyond the state of the art and open a new gateway to motivic homotopy theory.