Finiteness of cousin cohomologies

Abstract

The notion of the Cousin complex of a module was given by Sharp in 1969. It wasn’t known whether its cohomologies are finitely generated until recently. In 2001, Dibaei and Tousi showed that the Cousin cohomologies of a finitely generated A A -module  M M are finitely generated if the base ring  A A is local, has a dualizing complex, M M satisfies Serre’s ( S 2 ) (S_2) -condition and is equidimensional. In the present article, the author improves their result. He shows that the Cousin cohomologies of  M M are finitely generated if A A is universally catenary, all the formal fibers of all the localizations of  A A are Cohen-Macaulay, the Cohen-Macaulay locus of each finitely generated A A -algebra is open and all the localizations of  M M are equidimensional. As a consequence of this, he gives a necessary and sufficient condition for a Noetherian ring to have an arithmetic Macaulayfication.