  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.
