Relatively endotrivial complexes

Abstract

Let G be a finite group and k be a field of characteristic $p>0$. In prior work, we studied endotrivial complexes, the invertible objects of the bounded homotopy category of p-permutation kG-modules $K^b\left({}_{kG}\mathrm{triv}\right)$. Using the notion of projectivity relative to a kG-module, we expand on this study by defining notions of “relatively” endotrivial chain complexes, analogous to Lassueur's construction of relatively endotrivial kG-modules. We obtain equivalent characterizations of relative endotriviality and find corresponding local homological data which almost completely determine the isomorphism class of a relatively endotrivial complex. We show this local data must partially satisfy the Borel-Smith conditions, and consider the behavior of restriction to subgroups containing Sylow p-subgroups S of G.