On the equality between homological and cohomological dimension of groups. (Q2879867)

The authors present two criteria for a flat module over a ring to be projective. The first one is based on the relation between flatness and projectivity (see \textit{M. Raynaud} and \textit{L. Gruson} [Invent. Math. 13, 1-89 (1971; Zbl 0227.14010)]) and has as consequence a criterion for the equality \(\mathrm{hd}_{\mathbb Z}G=\mathrm{cd}_{\mathbb Z}G\) where \(G\) is any countable group. This result complements the structure theorem for soluble groups \(G\) with \(\mathrm{hd}_{\mathbb Z}G=\mathrm{cd}_{\mathbb Z}G<+\infty\).NEWLINENEWLINE The second criterion, based on the work of \textit{W. Dicks} and \textit{P. A. Linnell} [Math. Ann. 337, No. 4, 855-874 (2007; Zbl 1190.20021)]; \textit{P. Kropholler, P. Linnell} and \textit{W. Lück} [Lond. Math. Soc. Lect. Note Ser. 358, 272-277 (2009; Zbl 1211.18014)] has as application that a group \(G\) such that \(\mathrm{hd}_{\mathbb Z}G=1\) is locally free if the augmentation ideal \(I_ G\) of \(\mathbb ZG\) is residually nilpotent (i.e. \(\bigcap_{n=1}^{+\infty}I_G^n=0\)).