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# Homological Aspects of Semigroup Gradings on Rings and Algebras

Published:1999-06-01
Printed: Jun 1999
• W. D. Burgess
• Manuel Saorín
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## Abstract

This article studies algebras $R$ over a simple artinian ring $A$, presented by a quiver and relations and graded by a semigroup $\Sigma$. Suitable semigroups often arise from a presentation of $R$. Throughout, the algebras need not be finite dimensional. The graded $K_0$, along with the $\Sigma$-graded Cartan endomorphisms and Cartan matrices, is examined. It is used to study homological properties. A test is found for finiteness of the global dimension of a monomial algebra in terms of the invertibility of the Hilbert $\Sigma$-series in the associated path incidence ring. The rationality of the $\Sigma$-Euler characteristic, the Hilbert $\Sigma$-series and the Poincar\'e-Betti $\Sigma$-series is studied when $\Sigma$ is torsion-free commutative and $A$ is a division ring. These results are then applied to the classical series. Finally, we find new finite dimensional algebras for which the strong no loops conjecture holds.
 MSC Classifications: 16W50 - Graded rings and modules 16E20 - Grothendieck groups, $K$-theory, etc. [See also 18F30, 19Axx, 19D50] 16G20 - Representations of quivers and partially ordered sets

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