COHOMOLOGY OF SPLIT ALGEBRAS AND OF TRIVIAL EXTENSIONS

We consider associative algebras (Lambda) over a field provided with a direct sum decomposition of a two-sided ideal M and a sub-algebra A – examples are provided by trivial extensions or triangular type matrix algebras. In this relative and split setting we describe a long exact sequence computing the Hochschild cohomology of (Lambda). We study the connecting homomorphism using the cup-product and we infer several results, in particular the first Hochschild cohomology group of a trivial extension never vanishes.