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Cohen-Macaulayness in graded algebras. (English) Zbl 0844.13006
Cohen-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.

##### MSC:
 13C14 Cohen-Macaulay modules 13A02 Graded rings 14B05 Singularities in algebraic geometry 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
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