How to get $I_i \in add(\nu_A(Q))$ for $1 \leq i \leq n$ by $Ext^{i}_A(S,S)=0$?

Question

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 $Ext^{i}_A(S,S)=0$?

Answer 1

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).