  In this paper, we introduce the notion of $\nu$-stable silting-discrete
algebras, which unify silting-discrete algebras and tilting-discrete
self-injective algebras, where $\nu$ is a triangle auto-equivalence of the
bounded homotopy category of finitely generated projective modules. Moreover,
we give an example of tilting-discrete self-injective algebras but not
silting-discrete.
