Finitistic dimensions of finite dimensional monomial algebras

The geometry of uniserial representations of finite dimensional algebras. III: Finite uniserial type

“…One of the most interesting aspects of the following theorem lies in the strong restrictions which finiteness of the uniserial type of Λ imposes on the 'ground plans' for the composition series of uniserial Λ-modules (see (4) and (4') below). In rough terms, condition (4') says that, whenever a composition factor of a uniserial module occurs with multiplicity larger than 1, a whole segment of the composition series needs to repeat.…”

Finitistic dimension and restricted injective dimension

“…P r o o f. By [6], for any arbitrary finite numbers m and n, there is a finite dimensional algebra A with findim( A A) = Findim( A A) = m and findim(A A ) = Findim(A A ) = n. It is well known that findim( A A) = fin.inj.dim(A A ) and Findim( A A) = Fin.inj.dim(A A ). Let us take m > 0 and n = 0.…”

Infinite dimensional tilting modules over finite dimensional algebras

“…The first of these conjectures was disproved by in [41], and the second one has been proven for (i) monomial algebras in [36] (again in [44]), (ii) radical cube zero (and even more generally for algebras with Loewy length 2n q 1 and ΛT r n of finite representation type, see [32]) in [37] ( [43]), (iii) when P I mod Λ¡ is contravariantly finite in [9], (iv) when the representation dimension of Λ is at most 3 in [43] (all special biserial algebras has representation dimension at most 3, see [33]). None of the above proofs of the finiteness of findim Λ directly involve a tilting or a cotilting module.…”