Homological aspects of semigroup gradings on rings and algebras

This article studies algebras R over a simple artinian ring A, presented by a quiver and relations and graded by a semigroup Sigma. Suitable semigroup s often arise from a presentation of R. Throughout, the algebras need not b e finite dimensional. The graded K-0, along with the Sigma-graded Cartan en domorphisms and Cartan matrices, is examined. It is used to study homologic al properties. A test is found for finiteness of the global dimension of a monomial algebr a in terms of the invertibility of the Hilbert Sigma-series in the associat ed path incidence ring. The rationality of the Sigma-Euler characteristic, the Hilbert Sigma-series and the Poincare-Betti C-series is studied when Sigma is torsion-free comm utative and A is a division ring. These results are then applied to the cla ssical series. Finally, we find new finite dimensional algebras for which t he strong no loops conjecture holds.