Geodesic
On the Cohomological Dimension of Soluble Groups
Canadian mathematical bulletin, Tome 24 (1981) no. 4, pp. 385-392

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DOI

It is known that every torsion-free soluble group G of finite Hirsch number hG is countable, and its homological and cohomological dimensions over the integers and rationals satisfy the inequalities We prove that G must be finitely generated if the equality hG = cdQG holds. Moreover, we show that if G is a countable soluble group of finite Hirsch number, but not necessarily torsion-free, and if hG = cdQG, then hḠ = cdQḠ for every homomorphic image Ḡ of G.
DOI : 10.4153/CMB-1981-060-x
Mots-clés : 0014, 0015, soluble groups, Hirsch numbers, homological and cohomological dimension 
Gildenhuys, D.; Strebel, R. On the Cohomological Dimension of Soluble Groups. Canadian mathematical bulletin, Tome 24 (1981) no. 4, pp. 385-392. doi: 10.4153/CMB-1981-060-x
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     title = {On the {Cohomological} {Dimension} of {Soluble} {Groups}},
     journal = {Canadian mathematical bulletin},
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     year = {1981},
     volume = {24},
     number = {4},
     doi = {10.4153/CMB-1981-060-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1981-060-x/}
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