Relatively endotrivial complexes

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 ( triv k G ) . 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 .