Hochschild cohomology of algebras arising from categories and from bounded quivers
Résumé
The main objective of this paper is to present a theory for computing the Hochschild cohomology of algebras built on a specific data, namely multi-extension algebras. The computation relies on cohomological functors evaluated on the data, and on the combinatorics of an ad hoc quiver. One-point extensions are occurrences of this theory, and Happel's long exact sequence is a particular case of the long exact sequence of cohomology that we obtain via the study of trajectories of the quiver. We introduce cohomology along paths, and we compute it under suitable Tor vanishing hypotheses. The cup product on Hochschild cohomology enables us to describe the connecting homomorphism of the long exact sequence. Multi-extension algebras built on the round trip quiver provide square matrix algebras which have two algebras on the diagonal and two bimodules on the corners. If the bimodules are projective, we show that a five-term exact sequences arises. If the bimodules are free of rank one, we provide a complete computation of the Hochschild cohomology. On the other hand, if the corner bimodules are projective without producing new cycles in the data, Hochschild cohomology is that of the product of the algebras on the diagonal for large enough degrees. The word \emph{algebra} always means an associative $k$-algebra over a field $k$; an algebra is not necessarily finite dimensional.