Going from cohomology to Hochschild cohomology

The relation between ‘ordinary’ cohomology, and Hochschild cohomology is investigated for quotients of quiver algebras which are either graded or finite-dimensional. It is shown that a resolution of the direct sum of the simple modules of the algebra can often be made two-sided, and yield a resolution of the algebra itself over its enveloping algebra. The multiplicative structures of the cohomologies are shown to be related by a spectral sequence of algebras converging to the Hochschild cohomology, whose first term is given by a tensor product of the algebra itself with its cohomology algebra.