# Projective dimensions of the terms in a minimal injective resolution of the regular module

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 $$i>1$$? (probably yes, but im too blind at the moment to construct an example. It necessarily has to have global dimension at least 3.)

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)

edit: The reason might be as follows when $$pd(I_0)=0$$:

We have $$0 \rightarrow A \rightarrow I_0 \rightarrow \Omega^{-1}(A) \rightarrow 0$$ and thus all indecomposable injective modules of projective dimension one appear in $$\Omega^{-1}(A)$$ and thus also in $$I_1$$. Now it is probably easy to see that they cant appear later again, but Im not sure why at the moment.