by Benson, D. J.

Let k be a commutative ring of coefficients and G be a finite group. Does there exist a flat k G-module which is projective as a k-module but not as a k G-module? We relate this question to the question of existence of a k-module which is flat and periodic but not projective. For either question to have a positive answer, it is at least necessary to have |k| ≥ ℵ_ω. There can be no such example if k is Noetherian of finite Krull dimension, or if k is perfect.

DOI: 10.1023/A:1009918813260
Print publication date: 9/1/1999
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by Vancliff, Michaela

A family of flat deformations of a commutative polynomial ring S on n generators is considered, where each deformation B is a twist of S by a semisimple, linear automorphism σ of ℙ
ⁿ⁻¹, such that a Poisson bracket is induced on S. We show that if the symplectic leaves associated with this Poisson structure are algebraic, then they are the orbits of an algebraic group G determined by the Poisson bracket. In this case, we prove that if σ is ‘generic enough’, then there is a natural one-to-one correspondence between the primitive ideals of B and the symplectic leaves if and only if σ has a representative in GL(ℂ
ⁿ
) which belongs to G. As an example, the results are applied to the coordinate ring
$$\mathcal{O}_q (M_2 )$$
of quantum 2 × 2 matrices which is not a twist of a polynomial ring, although it is a flat deformation of one; if q is not a root of unity, then there is a bijection between the primitive ideals of
$$\mathcal{O}_q (M_2 )$$
and the symplectic leaves.

DOI: 10.1023/A:1009914728281
Print publication date: 9/1/1999
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by Ara, Pere

We develop the theory of Morita equivalence for rings with involution, and we show the corresponding fundamental representation theorem. In order to allow applications to operator algebras, we work within the class of idempotent nondegenerate rings. We also prove that two commutative rings with involution are Morita *-equivalent if and only if they are *-isomorphic.

DOI: 10.1023/A:1009958527372
Print publication date: 9/1/1999
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by Enochs, Edgar E.; Jenda, Overtoun M. G.; Xu, Jinzhong

Before his death, Auslander announced that every finitely generated module over a local Gorenstein ring has a minimal Cohen–Macaulay approximation. Yoshimo extended Auslander’s result to local Cohen–Macaulay rings admitting a dualizing module.Over a local Gorenstein ring the finitely generated maximal Cohen–Macaulay modules are the finitely generated Gorenstein projective modules so in fact Auslander’s theorem says finitely generated modules over such rings have Gorenstein projective covers. We extend Auslander’s theorem by proving that over a local Cohen–Macaulay ring admitting a dualizing module all finitely generated modules of finite G-dimension (in Auslander’s sense) have a Gorenstein projective cover. Since all finitely generated modules over a Gorenstein ring have finite G-dimension, we recover Auslander’s theorem when R is Gorenstein.

DOI: 10.1023/A:1009998109625
Print publication date: 9/1/1999
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by Ksir, Amy E.

Let W be a Weyl group and P ⊂ W, a parabolic subgroup. In this paper, we give the decomposition of the permutation representation Ind
_P

^W
1 into irreducibles for each exceptional W and maximal parabolic P. We find that there is an ‘extra’ common irreducible component which appears for exceptional groups and not for classical groups. This work is motivated by the study of Prym varieties and integrable systems.

DOI: 10.1023/A:1009946123738
Print publication date: 9/1/1999
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by Brüstle, Thomas; Hille, Lutz; Ringel, Claus Michael; Röhrle, Gerhard

It is well known that the Auslander algebra of any representation finite algebra is quasi-hereditary. We consider the Auslander algebra A
_n
of k[T]/〈
ⁿ
〉 (here, k is a field, T a variable and n a natural number). We determine all Δ-filtered A
_n
-modules without self-extensions. They can be described purely combinatorially. Given any Δ-filtered module N, we show that there is (up to isomorphism) a unique Δ-filtered module M without self-extensions which has the same dimension vector. In the case where k is an infinite field, N is a degeneration of this module M. In particular, we see that in this case, the set of Δ-filtered modules with a fixed dimension vector is the closure of an open orbit (thus irreducible). As observed by Hille and Röhrle, the problem of describing all Δ-filtered A
_n
-modules is the same as that of describing the conjugacy classes of elements in the unipotent radical of a parabolic subgroup P of GL(m, k) under the action of P, thus we recover Richardson’s dense orbit theorem in this instance.

DOI: 10.1023/A:1009999006899
Print publication date: 9/1/1999
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by Năstăsescu, Constantin; Panaite, Florin; Van Oystaeyen, Freddy

We apply to Hopf algebras a construction from graded rings, called “the group ring of a graded ring”. By using this construction we study the transfer of properties between certain categories of relative Hopf modules. As another application, we obtain a Maschke-type theorem for a Galois extension over a semisimple Hopf algebra.

DOI: 10.1023/A:1009931309850
Print publication date: 9/1/1999
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