Directions for the Long Exact Cohomology Sequence in Moore Categories

A new method for realizing the first and second order cohomology groups of an internal abelian group in a Barr-exact category was introduced in [6] and [10]. The main role, in each level, is played by a direction functor. This approach can be generalized to any level n and produces a long exact cohomology sequence. By applying this method to Moore categories we show that they represent a good context for non-abelian cohomology, in particular the Baer Extension Theory. Introduction One of the goals in Homological Algebra is to form a long exact cohomology sequence of abelian groups, given a short exact sequence of modules over a ring R with identity. The starting point is the well known property of the additive functor HomR(X,−) : R-Mod → Ab of preserving left exact sequences. Intuitively, the following question is raised: given a short exact sequence of R-modules N ֌ P ։ Q, can we prolong the exact sequence of abelian groups 0 → HomR(X,N) → HomR(X,P ) → HomR(X,Q) (infinitely) to the right? The affirmative answer is given through equivalence classes of n-fold extensions equipped with the Baer sum. These abelian groups can be computed as cohomology groups and are used to construct a long exact cohomology sequence. The problem can be generalized to the following expression: Given a short exact sequence N // k //P p // //Q (1) in Ab(E), can the exact sequence of abelian groups 0 HomE(1, N) k∗ HomE(1, P ) p∗ HomE(1, Q) (2) be extended (infinitely) to the right? Received January 3, 2007. The author gratefully acknowledges financial support from FCT/Centre of Mathematics of the University of Coimbra.