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# Finite Cohen--Macaulay Type and Smooth Non-Commutative Schemes

Published:2008-04-01
Printed: Apr 2008
• Peter J\o rgensen
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## Abstract

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.
 Keywords: Artin--Schelter Cohen--Macaulay algebra, Artin--Schelter Gorenstein algebra, Auslander's theorem on finite Cohen--Macaulay type, Cohen--Macaulay ring, fully bounded Noetherian algebra, isolated singularity, maximal Cohen--Macaulay module, non-commutative
 MSC Classifications: 14A22 - Noncommutative algebraic geometry [See also 16S38] 16E65 - Homological conditions on rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) 16W50 - Graded rings and modules

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