Homological Aspects of Semigroup Gradings on Rings and Algebras
Printed: Jun 1999
W. D. Burgess
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
16W50 - Graded rings and modules
16E20 - Grothendieck groups, $K$-theory, etc. [See also 18F30, 19Axx, 19D50]
16G20 - Representations of quivers and partially ordered sets