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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.

We determine the Hochschild cohomology algebras of the square-free monomial complete intersections. In particular, we provide a formula for the cup product which gives the cohomology module an algebra structure and then we describe this structure in terms of generators and relations. In addition, we compute the Hilbert series of the Hochschild cohomology of these algebras., 17 pages

We show that the universal Gr\"obner basis and the Graver basis of a binomial edge ideal coincide. We provide a description for this basis set in terms of certain paths in the underlying graph. We conjecture a similar result for a parity binomial edge ideal and prove this conjecture for the case when the underlying graph is the complete graph., Comment: 12 pages, 1 figure, v2: added introduction, updated layout, fixed several typos, added examples, standardised notation, changed references

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.

Salvetti constructed a cellular space B-D for any Artin group A(D) defined by a Coxeter graph D. We show that B-D is an Eilenberg-Mac Lane space if B-D' is an Eilenberg-Mac Lane space for every subgraph D' of D involving no infinity-edges.

Forman's discrete Morse theory is studied from an algebraic viewpoint, and we show how this theory can be extended to chain complexes of modules over arbitrary rings. As applications we compute the homologies of a certain family of nilpotent Lie algebras, and show how the algebraic Morse theory can be used to derive the classical Anick resolution as well as a new two-sided Anick resolution.

The generating function of the Betti numbers of the Heisenberg Lie algebra over a field of characteristic 2 is calculated using discrete Morse theory., Comment: 2 pages

Let $k$ be a field, and $A$ be a polynomial algebra over $k$. Let $I\subseteq A$ be an ideal. We present a novel method for computing resolutions of $A/I$ over $A$. The method is a synthesis of Grobner basis techniques and homological perturbation theory. The examples in this paper were computed using computer algebra.

In this note, the cyclic homology of quiver algebras with quadratic monomial relations over a field are computed. This is done by using a mixed complex due to Cibils. An associated spectral sequence is considered, which converges to the cyclic homology. The E 1 -term of this is calculated with the aid of the author's earlier work on the Hochschild homology of these algebras. The E 2 -term is then calculated, and since all higher differentials in the spectral sequence vanishes, the homology spaces can be determined from it.

It is proved that for any set M of squarefree monomials in the variables x1,…,xn, the algebra A=k[x1,…,xn]/(M) is Golod if and only if the algebra B=E(x1,…,xn)/(M) is Golod, where E is the exterior algebra. This is proved by showing the equivalence of the extremality of the Poincaré series of A and B.