Cyclic Covers of Rings with Rational Singularities
نویسنده
چکیده
We examine some recent work of Phillip Griffith on étale covers and fibered products from the point of view of tight closure theory. While it is known that cyclic covers of Gorenstein rings with rational singularities are Cohen-Macaulay, we show this is not true in general in the absence of the Gorenstein hypothesis. Specifically, we show that the canonical cover of a Q-Gorenstein ring with rational singularities need not be Cohen-Macaulay.
منابع مشابه
Rings of Singularities
This paper is a slightly revised version of an introduction into singularity theory corresponding to a series of lectures given at the ``Advanced School and Conference on homological and geometrical methods in representation theory'' at the International Centre for Theoretical Physics (ICTP), Miramare - Trieste, Italy, 11-29 January 2010. We show how to associate to a triple of posit...
متن کاملGeometric Interpretation of Tight Closure and Test Ideals
We study tight closure and test ideals in rings of characteristic p 0 using resolution of singularities. The notions of F -rational and F regular rings are defined via tight closure, and they are known to correspond with rational and log terminal singularities, respectively. In this paper, we reformulate this correspondence by means of the notion of the test ideal, and generalize it to wider cl...
متن کاملOn generalizations of semiperfect and perfect rings
We call a ring $R$ right generalized semiperfect if every simple right $R$-module is an epimorphic image of a flat right $R$-module with small kernel, that is, every simple right $R$-module has a flat $B$-cover. We give some properties of such rings along with examples. We introduce flat strong covers as flat covers which are also flat $B$-covers and give characterizations of $A$-perfe...
متن کاملF-rational Rings Have Rational Singularities
It is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseu-dorational. A key point in the proof is a characterization of F-rational local rings as those Cohen-Macaulay local rings (R; m) in which the local cohomology module H d m (R) (where d is the dimension of R) have no submodules stable under...
متن کاملAutomorphism Groups of Schur Rings
In 1993, Muzychuk [18] showed that the rational Schur rings over a cyclic group Zn are in one-to-one correspondence with sublattices of the divisor lattice of n, or equivalently, with sublattices of the lattice of subgroups of Zn. This can easily be extended to show that for any finite group G, sublattices of the lattice of characteristic subgroups of G give rise to rational Schur rings over G ...
متن کامل