2019/02/01 by Diego Bravo, Bravo, Diego, Charles Paquette +1
Mathematics · #16E10 #16G20 #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:16E10 #msc:16G20
paper · pdf · doi:10.48550/arxiv.1902.00317
9 pages
arxiv created 2019/11/04 · arxiv updated 2019/11/05
In this note, we prove that if Λ is an Artin algebra with a simple module S of finite projective dimension, then the finiteness of the finitistic dimension of Λ implies that of (1-e)Λ(1-e) where e is the primitive idempotent supporting S. We derive some consequences of this. In particular, we recover a result of Green-Solberg-Psaroudakis: if Λ is the quotient of a path algebra by an admissible ideal I whose defining relations do not involve a certain arrow α, then the finitistic dimension of Λ is finite if and only if the finitistic dimension of Λ/ΛαΛ is finite.