Inertial blocks of finite groups over arbitrary fields (Q6632085)

Let \(G\) be a finite group and \(p \in \pi(G)\). An \(\mathcal{O}\)-block of \(G\) is a primitive central idempotent of the group algebra \(\mathcal{O}G\). Let \(b\) be an \(\mathcal{O}\)-block of \(G\), \(P\) a defect group of \(b\) and \(c\) the Brauer correspondent of \(b\) in \(N_{G}(P)\). Assuming that \(k\) is algebraically closed and that \(P\) is abelian, Broué's conjecture says that the block algebras \(\mathcal{O}Gb\) and \(\mathcal{O}N_{G}(P)c\) are splendidly equivalent. In this paper, the authors investigate the possibility of finding a general method to prove that Broué's conjecture may be extended from algebraically closed fields towards over arbitrary fields.\N\NAn \(\mathcal{O}\)-block \(b\) of \(G\) is inertial if there is a Morita equivalence between the block algebras \(\mathcal{O}Gb\) and \(\mathcal{O}N_{G}(P)c\) induced by a bimodule with an endo-permutation source, where \(P\) is a defect group of \(b\) and \(c\) is the Brauer correspondent of \(b\) in \(N_{G}(P)\). When \(k\) is algebraically closed, such blocks are defined in [\textit{L. Puig}, Math. Z. 269, No. 1-2, 115--136 (2011; Zbl 1241.20012)]. An extension \(\mathcal{O}'\) of \(\mathcal{O}\) is a complete discrete valuation ring containing \(\mathcal{O}\) such that\N\N\(J(\mathcal{O})\subseteq J(\mathcal{O}')\), where \(J(\mathcal{O})\) and \(J(\mathcal{O}')\) are maximal ideals of \(\mathcal{O}\) and \(\mathcal{O}'\). An \(\mathcal{O}'\)-block \(b'\) of \(G\) is said to cover an \(\mathcal{O}\)-block \(b\) of \(G\) if \(bb'=b'\) in \(\mathcal{O}'G\).\N\NThe main result of the paper under review is Theorem 1.1: Let \(G\) be a finite group and \(b\) an \(\mathcal{O}\)-block of \(G\). Let \(\mathcal{O}'\) be an extension of \(\mathcal{O}\) and \(b'\) an \(\mathcal{O}'\)-block covering \(b\). Assume that \(p\) is odd or that \(P\) is abelian. Then \(b\) is inertial if and only if \(b'\) is inertial.