 
Summary: THE MOTIVIC NEARBY CYCLES AND THE CONSERVATION
CONJECTURE
JOSEPH AYOUB
To Jacob Murre for his 75th birthday
Contents
1. Introduction 1
2. The classical pictures 2
3. Specialization systems 6
4. Constructing the vanishing cycles formalism 11
5. Conservation conjecture. Application to Schur finiteness of motives 33
References 41
1. Introduction
Let X be a noetherian scheme. Following Morel and Voevodsky (see [24], [25], [28],
[33] and [37]), one can associate to X the motivic stable homotopy category SH(X).
Objects of SH(X) are Tspectra of simplicial sheaves on the smooth Nisnevich
site (Sm/X)Nis, where T is the pointed quotient sheaf A1
X/GmX. As in topology,
SH(X) is triangulated in a natural way. There is also a tensor product  X 
and an "internal hom": HomX on SH(X) (see [20] and [33]). Given a morphism
f : X // Y of noetherian schemes, there is a pair of adjoint functors (f
