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Let a and b be two coprime positive integers and k an arbitrary field. We determine the ring structure of the Hochschild cohomology of the numerical semigroup algebras k [ s a , s b ] of embedding dimension two (thus also complete intersections) in terms of generators and relations. In addition, we compute the Hilbert series of these cohomology rings.

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.