Let $A$ be a finite dimensional algebra with finite global dimension and with minimal injective coresolution $I_i$ of the regular module $A$.

The study of the projective dimensions of the $I_i$ is an important tool to test whether certain subcategories are extension closed or closed under submodules, see for example the article "Homolocial theory of noetherian rings" by Idun Reiten and https://www.sciencedirect.com/science/article/pii/0022404994900442 . 

Questions:

>1. Is there an easy example with $pd(I_i)=1$ for some $i>1$?

>2. Can we have  $pd(I_i)=1$ for some $i>1$ in case $pd(I_0)=0$?

>3. Can we have $pd(I_i)=1$ for some $i>1$ in case $A$ is a Nakayama algebra?

To my surprise my computer found no such example for a Nakayama algebra.

(of course this question has the danger that I oversee something obvious)