The ranks of homology of complexes of projective modules over finite groups (Q6590029)

Let \(G\) be a group and let \(p\) be a prime. An open conjecture of \textit{G. Carlsson} [Lect. Notes Math. 1217, 79--83 (1986; Zbl 0614.57023)] says that if \(G\) is an elementary abelian \(p\)-group of rank \(r\) acts freely on a finite CW-complex, then the sum of the dimensions of the homology groups is at least \(2r\). An algebraic version of the conjecture says that for such group \(G\), and for any finite dimensional complex of free \(kG\)-modules the sum of the dimensions of the homology groups of the complex is at least \(2r\).\N\N\textit{S. B. Iyengar} and \textit{M. E. Walker} [Acta Math. 221, No. 1, 143--158 (2018; Zbl 1403.13026)] gave a counter-example to this conjecture in the form of a cone over an endomorphism of the Koszul complex over \(kG\). In the paper under review, the author shows that this counter-example can be extended to examples over any finite group with many choices of the complex.