General notation, terminology and quoted results

Abstract

Throughout the paper p is a prime number, k a field of characteristic p, assumed algebraically closed except in Appendix 1, and O a complete discrete valuation ring with residue field k (we allow the possibility O = k). All the O-algebras we consider are associative and unitary except in Remark 11.12 below, but the homomorphisms between O-algebras are not required to be unitary. The unity elements are simply denoted by 1 whenever they have no previous name as idempotents in bigger O-algebras; abusively, 1 denotes also either the trivial element or the trivial subgroup in any group. All the O-modules and O-algebras we consider are assumed to be finitely generated as O-modules (in short O-finite) except when we mention explicitely the contrary; a significant exception is O-algebras ℱ and D that we introduce in Section 9 and employ in all the subsequent sections. If A is an O-algebra, we denote by Aut(A) the group of automorphisms of O, by A* the group of invertible elements of A, by J(A) the Jacobson radical of A, by Z(A) the center of A and by A° the opposite O-algebra; recall that

$$A = \sum\limits_{a \in {A^*}} {\mathcal{O}a} $$

(2.1.1)