Normal subgroups and support 𝜏-tilting modules

Abstract

Let \tilde{𝐺} be a finite group, 𝐺 a normal subgroup of \tilde{𝐺} and 𝑘 an algebraically closed field of characteristic 𝑝 > 0. The first main result in this paper is to show that support 𝜏-tilting 𝑘\tilde{𝐺}-modules with some properties are support 𝜏-tilting modules as 𝑘𝐺-modules, too. As the second main result, we give equivalent conditions for support 𝜏-tilting 𝑘\tilde{𝐺}-modules to satisfy the above properties, and show that the set of the support 𝜏-tilting 𝑘\tilde{𝐺}-modules with the properties is isomorphic to the set of \tilde{𝐺}-invariant support 𝜏-tilting 𝑘𝐺-modules as posets. As an application, we show that the set of \tilde{𝐺}-invariant support 𝜏-tilting 𝑘𝐺-modules is isomorphic to the set of support 𝜏-tilting 𝑘\tilde{𝐺}-modules in the case that the index of 𝐺 in \tilde{𝐺} is a 𝑝-power. As a further application, we give a feature of vertices of indecomposable 𝜏-rigid 𝑘\tilde{𝐺}-modules. Finally, we give block versions of the above results.