Homological Aspects of Semigroup Gradings on Rings and Algebras

Burgess, W. D. ; Saorín, Manuel

Canadian journal of mathematics, Tome 51 (1999) no. 3, pp. 488-505

Résumé

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.

Détail

DOI : 10.4153/CJM-1999-022-5

Mots-clés : 16W50, 16E20, 16G20

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