On Injective Modules and Support Varieties for the Small Quantum Group

Abstract

Let be the small quantum group at a root of unity associated to a simple Lie algebra ⁠. Generalizing a classical result for algebraic groups, we show that if M is a -module admitting a compatible torus action, then the injectivity of M can be detected by the restriction of M to certain root subalgebras of ⁠. We provide two proofs of this fact. The first is combinatorial and applies also to the higher Frobenius–Lusztig kernels of the big quantum group containing ⁠. The second proof is geometric and makes use of a new rank-variety-type result for the Borel subalgebras of ⁠. A brief application of the new rank variety result is also discussed.