Properties of AS-Cohen-Macaulay algebras

Abstract

An AS-Cohen-Macaulay algebra is the non-commutative graded analogue of a (commutative local) Cohen-Macaulay ring. This note will show how some central properties of commutative Cohen-Macaulay rings generalize to AS-Cohen-Macaulay algebras. We prove the following result. Theorem. An AS-Cohen-Macaulay algebra has a balanced dualizing complex if and only if it is a graded factor of an AS-Gorenstein algebra. Theorem. Let A be an FBN AS-Cohen-Macaulay algebra. Then • • A has an artinian ring of quotients. • • Every minimal prime ideal /op of A is graded, and GK dim( A /op ) = GK dim(A) . • • A has a balanced dualizing complex of the form K[n] far a bi-module K, and if x ϵ A is regular, then x is also regular on K (from both sides). Theorem. Let A be FBN, N-graded and connected. Then the following conditions are equivalent: • • The algebra A is AS-Cohen-Macaulay. • • We have depth A ( A) = GKdim( A). • • The algebra A satisfies the following “inequality of the grade”: for any X ϵ D fg b(GrMod(A)), we have the inequality grade A ( X) ≥ GK dim( A) — GK dim( X).