THE FIRST COHOMOLOGY GROUP OF AN ALGEBRA WITH COEFFICIENTS

                                                                
Claude CIBILS   Manuel SAORIN

 In this paper we provide formulas for the dimension of the first Hochschild 
cohomology vector space of an algebra with a separable splitting subalgebra 
and coefficients in a bimodule. We first consider a monomial non-commutative 
algebra, in order to provide a simpler specific formula in this case and to 
sketch the methods used in the sequel. Recall that such an algebra is the 
quotient of the path algebra of a quiver by a two-sided ideal generated by 
oriented paths of length at least 2.

The important point is that the quiver can afford oriented cycles. The 
computation is based on a modelization of cohomological tools and on a 
computation of the center of those algebras.

In Section 3, we tackle the most general situation, with different bimodules 
of coefficients and/or more general algebras.