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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.

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