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Homological Aspects of Semigroup Gradings on Rings and Algebras

Published online by Cambridge University Press:  20 November 2018

W. D. Burgess  and
Manuel Saorín
W. D. Burgess
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, K1N 6N5 email: [email protected]
Manuel Saorín
Affiliation:
Departemento de Matemáticas, Universidad de Murcia, 30100 Espinardo, Murcia, Spain email: [email protected]
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Abstract

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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 Poincaré-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.

Keywords

16W5016E2016G20
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 3 , 01 June 1999 , pp. 488 - 505
Copyright
Copyright © Canadian Mathematical Society 1999

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