Cohen-Macaulayness in graded algebras.

*(English)*Zbl 0844.13006Cohen-Macaulayness of Rees algebras and associated graded rings of ideals have been actively investigated in recent years. We illustrate here how some methods introduced by J. B. Sancho de Salas [in: Géométrie algébrique et applications, La Rabida 1984, Part I, Trav. Cours 22, 201-209 (1987; Zbl 0625.14025)] yield additional insights in this area. Theorem 4.1 relates the Cohen-Macaulay property of a Rees algebra to the vanishing of cohomology of its Proj. As a special case, we get that in a normal local ring \(R\), there exists an ideal \(I\) whose blow-up desingularizes \(R\) and whose Rees algebra \(R[It]\) is Cohen-Macaulay (CM) if and only if \(R\) has a rational singularity – at least in characteristic zero, and more generally if desingularization theorems such as have been announced by Spivakovsky hold up. [In contrast, theorem 4.3, due to J. B. Sancho de Salas (loc. cit.) implies that for any CM local ring on a variety over \(\mathbb{C}\), there exists a desingularizing \(I\) whose associated graded ring is CM.] Theorem 5 gives an affirmative answer to a question of Huneke: It states that if \(R\) is pseudo-rational and if \(I\) is a non-zero \(R\)-ideal whose associated graded ring is CM, then \(R[It]\) is CM, too.