On birational Macaulayfications and Cohen–Macaulay canonical modules

Abstract

The aim of the paper is twofold. At first there is a characterization of those local domains admitting a birational Macaulayfication. It turns out that a local domain A, quotient of a Gorenstein ring, possesses a Cohen–Macaulay birational extension B if and only if its canonical module K( A) is a Cohen–Macaulay module. In this situation it follows that B is isomorphic to the endomorphism ring of K( A). The second part of the paper is devoted to the question when the canonical module of a local ring is a Cohen–Macaulay module. There are applications on sequentially Cohen–Macaulay rings. A large class of rings whose canonical module are Cohen–Macaulay is given by the simplicial affine semigroup rings. As a technical tool there is a spectral sequence related to the duality with respect to the dualizing complex.