Summary: CHARACTERISTIC p ANALOGUE OF MODULES
WITH FINITE CRYSTALLINE HEIGHT
Abstract. In the case of local fields of positive characteristic we introduce an ana-
logue of Fontaine's concept of Galois modules with crystalline height h N. If h = 1
these modules appear as geometric points of Faltings's strict modules. We obtain
upper estimates for the largest upper ramification numbers of these modules and
prove (under an additional assumption) that these estimates are sharp.
Let p be a prime number. Let K be a complete discrete valuation field with
perfect residue field k of characteristic p. Choose a separable closure Ksep of K and
set K = Gal(Ksep/K). Denote by R the valuation ring of K and for any v > 0,
K the ramification subgroup of K with the upper number v.
Suppose, first, that K is of characteristic 0, i.e. K contains Qp, and consider e =
e(K) -- the ramification index of K over Qp. In this situation for h N, Fontaine
[Fo3] introduced the category of finite Zp[K]-modules with crystalline height h.
Examples of such modules are given by subquotients of crystalline representations
of K with Hodge-Tate filtration of length h or, more specifically, of Galois modules