Tensor Hochschild homology and cohomology
Claude Cibils
D\'ep. de math\'ematiques, Universit\'e de Montpellier 2  

Interactions between ring theory and representations of algebras (Murcia), 
35--51, Lecture Notes in Pure and Appl. Math., 210, Dekker, New York, 2000. 

Abstract.
We consider the non commutative setting given by a ring A, an A-bimodule M 
and T the corresponding tensor algebra. We prove that the Hochschild 
cohomology of quotients T/I by positive ideals coincides with the homology 
of A whenever the quiver of M has no oriented cycles.  

If the quiver is an arrow (i.e. T is a triangular two by two matrix algebra) 
the Hochschild cohomology belongs to a long exact sequence. 
For other quivers, a spectral sequence converging to the Hochschild 
cohomology will be described in a forthcoming paper.