Distributivity in lattices of torsion classes over finite-dimensional algebras

Let [Formula: see text] be a basic finite-dimensional algebra and denote by [Formula: see text] the collection of all torsion classes of [Formula: see text]. It has been proved in [L. Demonet, O. Iyama, N. Reading, I. Reiten and H. Thomas, Lattice theory of torsion classes: Beyond [Formula: see text]-tilting theory, Trans. Amer. Math. Soc. Ser. B 10 (2023) 542–612. https://doi.org/10.1090/btran/100 ] that tors [Formula: see text] is always a completely semidistributive lattice. In this paper, we investigate the distributivity of this lattice and prove that the lattice is distributive if and only if the algebra is the finite direct product of finite-dimensional local algebras.