Some Criteria for the Cohen-Macaulay Property and Local Cohomology

Let a be an ideal of a commutative Noetherian ring R and M be a finitely generatedR-module of dimension d. We characterize Cohen-Macaulay rings in term of a special homologicaldimension. Lastly, we prove that if R is a complete local ring, then the Matlis dual of top localcohomology module H~d(M) is a Cohen-Macaulay R-module provided that the R-module M satisfiessome conditions.