  We define and study new filtrations called of stratification of a perverse
sheaf on a scheme; beside the cases of the weight or monodromy filtrations,
these filtrations are available whatever are the ring of coefficients. We
illustrate these constructions in the geometric situation of the simple unitary
Shimura varieties of Harris and Taylor's book for the perverse sheaves of
Harris-Taylor and the complex of vanishing cycles, introduced and studied in my
2009 paper at inventiones. In the situation studied in loc. cit., we show how
to use these filtrations to simplify the principal step of this paper; the
cases of finite field or ring of integer of a local field will be studied in
the next published paper.
