Igusa-Todorov ϕ Function for Truncated Path Algebras

Abstract

Given a truncated path algebra A = kQ J k we prove that φdimA = φdimA op . We also compute the φ-dimension of A in function of the φdimension of kQ J 2 when Q has no sources nor sinks. This allows us to bound the φ-dimension for truncated path algebras. Finally, we characterize A when its φ-dimension is equal to 1.

A Counterexample to the $ϕ$-Dimension Conjecture

“…The classes of algebras that have been shown to satisfy the finitistic dimension conjecture using the φ and ψ functions include: algebras of representation dimension at most 3 [IT05], algebras of finite injective dimension [LM18], Gorenstein algebras [LM18], truncated path algebras [BMR19], monomial relation algebras [LM18], Igusa-Todorov algebras [Wei09], and, indirectly, special biserial algebras [EHIS04].…”