Rigid and $\tau$-locally free modules via Taft algebras

We show that rigid modules are $\tau$-locally free for a large family of 1-Iwanaga–Gorenstein algebras. This family includes many cases of the algebras introduced by Geiss–Leclerc–Schröer in their Inventiones paper “Quivers with relations for symmetrizable Cartan matrices I: Foundations”, hence answering a question posed by Geiss–Leclerc–Schröer for many new cases. Our result is achieved by introducing coactions of Taft algebras and applying the recent theory developed by the first-named author and Coulembier.