Vanishing of $Ext^i_R(N, R)$

2012-08-13T06:23:44Z

This question is related to http://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 $i > d$?

Answer by Graham Leuschke for Vanishing of $Ext^i_R(N, R)$

2012-09-11T20:21:29Z

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-Sega (journal link): 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.

Vanishing of $Ext^i_R(N, R)$ - MathOverflow

Vanishing of $Ext^i_R(N, R)$

Answer by Graham Leuschke for Vanishing of $Ext^i_R(N, R)$