by Enochs, Edgar; Estrada, Sergio; Torrecillas, Blas

Every module over an Iwanaga–Gorenstein ring has a Gorenstein flat cover [13] (however, only a few nontrivial examples are known). Integral group rings over polycyclic-by-finite groups are Iwanaga–Gorenstein [10] and so their modules have such covers. In particular, modules over integral group rings of finite groups have these covers. In this article we initiate a study of these covers over these group rings. To do so we study the so-called Gorenstein cotorsion modules, i.e. the modules that split under Gorenstein flat modules. When the ring is ℤ, these are just the usual cotorsion modules. Harrison [16] gave a complete characterization of torsion free cotorsion ℤ-modules. We show that with appropriate modifications Harrison's results carry over to integral group rings ℤG when G is finite. So we classify the Gorenstein cotorsion modules which are also Gorenstein flat over these ℤG. Using these results we classify modules that can be the kernels of Gorenstein flat covers of integral group rings of finite groups. In so doing we necessarily give examples of such covers. We use the tools we develop to associate an integer invariant n with every finite group G and prime p. We show 1≤n≤|G : P| where P is a Sylow p-subgroup of G and gives some indication of the significance of this invariant. We also use the results of the paper to describe the co-Galois groups associated to the Gorenstein flat cover of a ℤG-module.

DOI: 10.1007/s10468-005-0339-2
Print publication date: 10/1/2005
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by Reineke, Markus

Noncommutative Hilbert schemes, introduced by M. V. Nori, parametrize left ideals of finite codimension in free algebras. More generally, parameter spaces of finite-codimensional submodules of free modules over free algebras are considered. Cell decompositions of these varieties are constructed, whose cells are parametrized by certain types of forests. Asymptotics for the corresponding Poincaré polynomials and properties of their generating functions are discussed.

DOI: 10.1007/s10468-005-8762-y
Print publication date: 10/1/2005
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by Böhm, Gabriella

The theory of integrals is used to analyze the structure of Hopf algebroids. We prove that the total algebra of a Hopf algebroid is a separable extension of the base algebra if and only if it is a semi-simple extension and if and only if the Hopf algebroid possesses a normalized integral. It is a Frobenius extension if and only if the Hopf algebroid possesses a nondegenerate integral. We give also a sufficient and necessary condition in terms of integrals, under which it is a quasi-Frobenius extension, and illustrate by an example that this condition does not hold true in general. Our results are generalizations of classical results on Hopf algebras.

DOI: 10.1007/s10468-005-8760-0
Print publication date: 10/1/2005
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by Lewis, Mark L.

Let G be a finite group, and write cd(G) for the set of degrees of irreducible characters of G. We say G satisfies the one-prime hypothesis if whenever a and b are distinct degrees in cd(G), then the greatest common divisor of a and b is either 1 or a prime. We show that if G is a solvable group satisfying the one-prime hypothesis, then |cd(G)|≤9. We also construct a solvable group G satisfying the one-prime hypothesis with |cd(G)|=9 which shows that the bound found in this paper is the best possible bound.

DOI: 10.1007/s10468-005-3596-1
Print publication date: 10/1/2005
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by Garkusha, Grigory; Prest, Mike

The relationship between the Ziegler spectrum of (the category of modules over) a ring and the Ziegler spectrum of its derived category is investigated. Over von Neumann regular rings and hereditary rings the spectrum of the derived category is a disjoint union of copies of the spectrum of the ring but in general there are further indecomposable pure-injective objects of the derived category.

DOI: 10.1007/s10468-005-8147-2
Print publication date: 10/1/2005
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by Iyama, Osamu

We study Nakayama pairs in τ-categories, which are pairs of objects connected by a certain diagram of Auslander–Reiten sequences. Using them, we naturally introduce orderlikeτ-categories. Then we study rejective subcategories of τ-categories. A typical example of orderlike τ-categories is given by the category of lattices over an order. In this case, its rejective subcategories correspond bijectively to its overrings.

DOI: 10.1007/s10468-005-0969-4
Print publication date: 10/1/2005
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