Gorenstein Flat Covers and Gorenstein Cotorsion Modules Over Integral Group Rings |
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Authors: | Email author" target="_blank">Edgar?EnochsEmail author Sergio?Estrada Blas?Torrecillas |
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Institution: | (1) Department of Mathematics, University of Kentucky, Lexington, KY, 40506-0027, U.S.A.;(2) Departamento de álgebra, Universidad de Granada, C El Greco s/n, 51002 Ceuta, Spain;(3) Departamento de álgebra y Análisis Matemático, Universidad de Almería, 04071 Almería, Spain |
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Abstract: | 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.
Presented by A. Verschoren
Mathematics Subject Classifications (2000) 20C05, 16E65. |
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Keywords: | Gorenstein flat and cotorsion modules over ℤ G Gorenstein flat cover co-Galois group |
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