Let $R$ be a commutative Noetherian ring and $I$ is a proper ideal of $R$. suppose that $M$ is a f.g. $R$-module.


$\DeclareMathOperator\Ext{Ext}$I'm looking for an example that has this property:


$$\Ext^i\_R(M,R/I)\neq 0$$ for all $i$ and there exists $n$ such that for all $i>n$


$$\Ext^i\_R(R/I,R/I)=0.$$


I don't know how to deal with this question. Any help will be appreciated.



Example of non vanishing Ext
Take $R$ a local artinian non-Gorenstein ring, $I=0$, $M$ the residue field of $R$. The vanishing of $Ext^i\_R(R/I,R/I)=Ext^i\_R(R,R)$ for $i>0(=:n)$ is immediate, while for the non-vanishing of $Ext^i\_R(M,R/I)=Ext^i\_R(k,R)$ see e.g. Bourbaki, Algèbre Commutative, chap. X, $\S$ 3, n. 7, Lemme 2.


