Logarithmic prismatic cohomology, motivic sheaves, and comparison theorems
Abstract
We prove that (logarithmic) prismatic and (logarithmic) syntomic cohomology are representable in the category of logarithmic motives. As an application, we obtain Gysin maps for prismatic and syntomic cohomology, and we explicitly identify their cofibers. We also prove a smooth blowup formula and we compute prismatic and syntomic cohomology of Grassmannians. In the second part of the paper, we develop a descent technique inspired by the work of Nizioł~ on log $K$theory. Using the resulting \emph{saturated descent}, we prove de Rham and crystalline comparison theorems for log prismatic cohomology, and the existence of Gysin maps for $A_{\inf}$cohomology.
 Publication:

arXiv eprints
 Pub Date:
 December 2023
 DOI:
 10.48550/arXiv.2312.13129
 arXiv:
 arXiv:2312.13129
 Bibcode:
 2023arXiv231213129B
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Algebraic Topology;
 Mathematics  KTheory and Homology;
 Mathematics  Number Theory;
 14F30 (Primary) 14F42;
 14A21;
 13D03 (Secondary)
 EPrint:
 51 pages, added smooth blowup formula and computations of the cohomology of Grassmanians. A few typos corrected