On some cohomological invariants for large families of infinite groups
In this article, the author investigates some cohomological invariants of the class of infinite groups denoted \(LH\mathcal F_{\phi,A}\). This class is defined using the groups of type \(\Phi\), introduced by Talelli over \(\mathbb Z\) in [\textit{O. Talelli}, Arch. Math. 89, No. 1, 24--32 (2007; Zbl 1158.20026)], and groups derived from a hierarchy of groups introduced by \textit{P. H. Kropholler} [J. Pure Appl. Algebra 90, No. 1, 55--67 (1993; Zbl 0816.20042)]. In the article under review, the author extends some results that were known over the ring of integers to arbitrary commutative rings of finite global dimension. In particular, one of the main result states that if a group \(\Gamma\) is in \(LH\mathcal F_{\phi,A}\) with \(A\) of finite global dimension, say \(t\), then the projective dimension of the module of bounded functions \(B(\Gamma,A)\), of \(\Gamma\) on \(A\) over \(A\Gamma\), is equal to the Gorenstein cohomological dimension of \(\Gamma\) over \(A\). Moreover, this quantity, say \(\theta\), is bounded above by the finistic dimension of \(A\Gamma\), which is itself bounded above by \(\theta+t\).
scientific article; zbMATH DE number 7382052
scientific article