A commutative local Cohen--Macaulay ring $R$ of finite Cohen--Macaulay type is known to be an isolated singularity; that is, $\Spec(R) \setminus \{ \mathfrak {m} \}$ is smooth. This paper proves a non-commutative analogue. Namely, if $A$ is a (non-commutative) graded Artin--Schelter \CM\ algebra which is fully bounded Noetherian and has finite Cohen--Macaulay type, then the non-commutative projective scheme determined by $A$ is smooth.

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.