# Gysin triangles in the category of motifs with modulus

## TL;DR

This paper explores Gysin triangles within the category of motives with modulus, linking them to log-crystalline cohomology and comparing motives with modulus to Voevodsky motives, revealing a decomposition into p-torsion and Voevodsky parts.

## Contribution

It introduces a Gysin triangle in motives with modulus as a motivic lift of log-crystalline Gysin triangles and compares motives with modulus to Voevodsky motives.

## Key findings

- Decomposition of motives with modulus into p-torsion and Voevodsky parts
- Motivic analogue of the relationship between rigid and log-crystalline cohomology
- Establishment of a Gysin triangle in motives with modulus

## Abstract

In this paper, we study a Gysin triangle in the category of motives with modulus. We can understand this Gysin triangle as a motivic lift of the Gysin triangle of log-crystalline cohomology due to Nakkajima and Shiho. After that we compare motives with modulus and Voevodsky motives. The corollary implies that an object in the category of motives with modulus decomposes into a $p$-torsion part and a Voevodsky motive part. We can understand the corollary as a motivic analogue of the relationship between rigid cohomology and log-crystalline cohomology.