Injective Dimensions And Almost Split Sequences

Let R be a commutative artin ring, and A is an artin algebra over R. Let δ:0â†'Aâ†'Bâ†'Câ†'0be an almost split sequence in mod-Λ, then we have id B<max{id A, id C}. In this paper we will discuss when id B<max{id A, id C} holds.This article is divided into two parts.In the first part, we give the preliminary knowledge. We first introduce definitions and relative properties of almost split sequences and irreducible morphisms. Then we prove some lemmas in terms of contravariant defect functor.In the second part, we give the definition of inequality sequences. Al-so we show a series of results related with the inequality of almost split sequences. In particular, We give a characterization of the inequality of almost split sequences. The characterization is given in terms of five condi-tions which C has to satisfy. Finally, we discuss the number of almost split sequences with inequality in mod-A, and then give some other important conclusions.