$\DeclareMathOperator\Ext{Ext}$This question is related to [Are the supports of $Ext^i(M,N)$ eventually periodic?](https://mathoverflow.net/questions/32310/are-the-supports-of-extim-n-eventually-periodic) .


Let $(R, \mathfrak{m}, k)$ be a Noetherian local ring of dimension $d$. It well known that


**Theorem** $R$ is Gorenstein iff $\Ext^i\_R(k, R) = 0$ for some $i > d$, and iff $\Ext^i\_R(k, R) = 0$ for all $i > d$.


By above Theorem we have: $\Ext^i\_R(k, R) = 0$ for some $i > d$ iff $\Ext^i\_R(k, R) = 0$ for all $i > d$.


The following question consider in a generalization of this situation for modules of finite length.


**Question:** Let $(R, \mathfrak{m}, k)$ be a Noetherian local ring of dimension $d$, $N$ an $R$-module of finite length. Is it true that: $\Ext^i\_R(N, R) = 0$ for some $i > d$ iff $\Ext^i\_R(N, R) = 0$ for all $i > d$?



Vanishing of $\mathrm{Ext}^i_R(N, R)$
No.


Let's work with an Artinian local ring $R$. Let $X$ be a module satisfying $\mathrm{Ext}\_R^i(X,R)\neq 0$ for all $i$. (These are plentiful; for example, assume $R$ is non-Gorenstein and let $X$ be the residue field.) Define a module $Q$ by "Serre's trick": take generators $\chi\_1, \dots, \chi\_r$ for $\mathrm{Ext}\_R^1(X,R)$ and consider the short exact sequence corresponding to $(\chi\_1, \dots, \chi\_r) \in \mathrm{Ext}\_R^1(X,R^r)$:
$$0 \to R^r \to Q \to X \to 0$$
Then $\mathrm{Ext}\_R^1(Q,R) =0$, since the long exact sequence of $\mathrm{Ext}$ looks like
$$\cdots \to \mathrm{Hom}\_R(R^r,R) \to \mathrm{Ext}\_R^1(X,R) \to \mathrm{Ext}\_R^1(Q,R) \to \mathrm{Ext}\_R^1(R^r,R)=0$$
and the map $\mathrm{Hom}\_R(R^r,R) \to \mathrm{Ext}\_R^1(X,R)$ is cooked up precisely to be surjective. On the other hand, $\mathrm{Ext}\_R^i(Q,R) = \mathrm{Ext}\_R^{i}(X,R) \neq 0$ for all $i\geq 2$.


One can jazz this up a bit, using a result of [Jorgensen–Şega](https://mathscinet.ams.org/mathscinet-getitem?mr=2238367) ([journal link](https://doi.org/10.1007/s10468-005-0559-5)): There exists a local Artinian ring $R$ and a family $\{M\_s\}\_{s\geq 1}$ of reflexive $R$-modules such that (among other things) $\mathrm{Ext}\_R^i(M\_s,R)\neq 0$ if and only if $1 \leq i \leq s-1$. They give a completely explicit construction of $R$ and the modules $M\_s$. Taking one of the $M\_s$ in place of $X$ above, one obtains modules $Q\_{a,b}$ for which $\mathrm{Ext}\_R^i(Q\_{a,b},R)$ vanishes up to $i=a-1$, is nonzero for $a \leq i \leq b$, and vanishes again for $i \geq b+1$. Then taking direct sums gives essentially arbitrary behavior of vanishing and non-vanishing.


