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. |