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Information Center for Mathematical Science

논문검색

Information Center for Mathematical Science

논문검색

Canadian Journal of Mathematics
( Vol. 51 NO.3 / (1999))
Homological Aspects of Semigroup Gradings on Rings and Algebras
W. D. Burgess, Manuel Saorin,
Pages. 488-505
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.
indent 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.
indent The rationality of the $Sigma$-Euler characteristic, the Hilbert $Sigma$-series and the Poincare-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.
Contents Introduction
1. Admissible gradings on rings
2. Graded Cartan matrices and graded Cartan endomorphisms
3. Applications to monomial algebras
4. On the rationality of the Hilbert $Sigma$-series
5. A remark on the ""strong no loops conjecture""
Key words
Mathmatical Subject Classification 16W50, 16E20, 16G20