# Question on $n$-regular modules

Let $A$ be finite dimensional connected algebra. A simple module $S$ is called $n$-regular in case $pd(S)=n$, $Ext_A^i(S,A)=0$ for $i=0,1,...,n-1$ and $Ext_A^n(S,A)$ being a simple $A$-left module. See for example https://www.sciencedirect.com/science/article/pii/S0001870818302809 for the relvance of 2-regular simple modules. We can assume $A$ is a quiver algebra and then being $n$-regular simply means that the injective envelope $I(S)$ of $S$ (having projective dimension $n$) occurs uniquely in the minimal injective coresolution $(I_i)$ of $A$ as a summand of $I_n$.

Questions:

1. In case every simple module of projective dimension $n$ (and there exists at least one such simple module) is $n$-regular, does $A$ have global dimension $n$? (In case this is false, is it true when assuming $A$ hsa finite global dimension?)

2. When every right simple module of projective dimension $n$ is $n$-regular (and there exists at least one such simple module) is the same true for every left simple modules of projective dimension $n$?

• Sorry, my favourite piece of pedantry: at least for the first question I presume you want $A$ to be connected? – Jeremy Rickard Sep 13 at 17:13
• @JeremyRickard Yes, thanks I added this condition. – Mare Sep 13 at 17:49