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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.

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    $\begingroup$ What’s the triple ext? $\endgroup$
    – Fernando Muro
    Commented Feb 1, 2022 at 19:37
  • $\begingroup$ @FernandoMuro I edited the formula, thanks. $\endgroup$
    – pink floyd
    Commented Feb 1, 2022 at 19:51

1 Answer 1

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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.

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    $\begingroup$ what is your mean of A? $\endgroup$
    – pink floyd
    Commented Feb 5, 2022 at 6:31

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