Quasi-Frobenius Rings and Generalizations QF-3 and QF-1 Rings

Auslander–Reiten theory revisited

“…Let n and m be positive integers. Following Tachikawa [115] and Hoshino [73], we write dom. dim Γ ≥ n if I i is a flat Γ-module for any 0 ≤ i < n. More generally, we say that Γ satisfies the (m, n)-condition [83,74] if fd I i < m holds for any 0 ≤ i < n. Notice that when Γ is a finite-dimensional algebra, we have fd I i = pd I i .…”

Generalized Auslander-Reiten conjecture and tilting equivalences

“…However, for an Artin algebra satisfying the finitistic dimension conjecture, we don't know if it also satisfies the Auslander-Reiten conjecture. In fact, if we restrict R to be self-injective, then the Auslander-Reiten conjecture is just the Tachikawa conjecture [7], which is still open up to now.…”

Loewy coincident algebra and QF-3 associated graded algebra

“…For the definition of QF -3 algebras, see Thrall [6] and Tachikawa [4]. Note that Auslander algebras are a special class of QF -3 algebras; recall that an Auslander algebra is the endomorphism algebra of the direct sum of all indecomposable modules over an algebra of finite representation type (Auslander [1]).…”