Let $A$ be a k-algebra, where k is a fixed field. Let $S$ be a simple, non-injective $A$-module such that $Ext^{i}\_{A}(S,S)=0$ for $1 \leq i \leq n$. Let $P(S)$ be the projective cover of $S$, and let $Q$ be the direct sum of all non-isomorphic indecomposable projective $A$-module which are not isomorphic to $P(S)$. $\nu\_A$ is the Nakayama functor.


Suppose $$0 \rightarrow S \rightarrow I(S) \rightarrow I\_1 \rightarrow \cdots \rightarrow I\_n \rightarrow I\_{n+1} \rightarrow \cdots$$ is a minimal injective resolution of $S$, where $I(S)$ is the injective cover of $S$. Who can help me get $I\_i \in add(\nu\_A(Q))$ for $1 \leq i \leq n$ by $Ext^{i}\_A(S,S)=0$?



How to get $I_i \in add(\nu_A(Q))$ for $1 \leq i \leq n$ by $Ext^{i}_A(S,S)=0$?
You just need the following fact for any simple module $S$ and any other module $M$: $Ext^{i}(S,M)=0$ iff in the minimal injective resolution $(I\_i)$ of $M$, $S$ is not a submodule of $I\_i$. You can find this in the first volume of Bensons book "Representations and Cohomology", somehwere in the beginning (have not the book at hand atm). 


