Gorenstein dimension and group cohomology with group ring coefficients

The Gorenstein cohomological dimension of a group G generalizes the ordinary cohomological dimension of G, in the sense that the two invariants coincide when the latter one is finite. In this paper, we show that the Gorenstein cohomological dimension GcdkG of G over a commutative ring k shares many...

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Bibliographic Details
Published in:Journal of the London Mathematical Society 2018-04, Vol.97 (2), p.306-324
Main Authors: Emmanouil, Ioannis, Talelli, Olympia
Format: Article
Language:English
Subjects:
18G20 (primary)
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Summary:The Gorenstein cohomological dimension of a group G generalizes the ordinary cohomological dimension of G, in the sense that the two invariants coincide when the latter one is finite. In this paper, we show that the Gorenstein cohomological dimension GcdkG of G over a commutative ring k shares many properties with the cohomological dimension of G over k. For example, if k has finite global dimension, the finiteness of GcdkG implies that GpdkGM is finite for any kG‐module M. Other properties concern the dependence of the Gorenstein cohomological dimension upon the coefficient ring, its behavior with respect to subgroups, the subadditivity with respect to extensions and the formula for the dimension of an ascending union. We also prove that if k has finite global dimension then the finiteness of GcdkG gives a criterion for the conditions cdkG
ISSN:0024-6107
1469-7750
DOI:10.1112/jlms.12103