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 $n$?

  • 1
    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

Your Answer

 
discard

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.