 
Summary: Motives of rigid varieties and
the motivic nearby functor
Joseph Ayoub
Venice, 22 june 2006
Constructing the category of motives
of rigid varieties
Let k be a complete field for a nonarchimedean
norm . : k R+. Denote RigV ar/k the
category of rigid varieties over k and RigSm/k
its subcategory of smooth varieties. One can
construct out of RigSm/k a triangulated
category RigDMeff(k) of rigid motives in the
same way as Voevodsky constructed the
category DMeff(k):
Step 1: Define an additive category RigCor(k)
with the same objects as RigSm/k
and morphisms RigCor(X, Y ) the free abelian
group on closed and irreducible subvarieties
Z X × Y which are finite and surjective over
a connected component of X. The composi
