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Let $R$ be a Noetherian, commutative ring and $M,N$ be finitely generated $R$-modules.

Question: Do the sets of minimal primes of $\text{Ext}^i_R(M,N)$ (for a fixed pair of $M,N$) become periodic for $i\gg 0$?

Background/Motivation: In this part I will assume $R$ local (I don't think it's significant for the question, but some statements will be awkward in general). I was motivated to ask after this recent question.

It is known that the asymptotic behavior of these $\text{Ext}$ is governed by the singularity of $R$. For instance, if $R$ is regular, then $\text{Ext}^i_R(M,N)=0$ for $i>\dim R$ because $R$ has finite global dimension equals to $\dim R$.

If $R$ is a hypersurface, then it follows from Eisenbud's original paper on matrix factorization that the minimal resolution of $M$ becomes eventually periodic of period $2$ after $\dim R$ spots, so the answer in this case is also affirmative.

With a little bit more work, one can show that the supports of $\text{Ext}^i_R(M,N)$ (in fact, the annihilators) become eventually periodic of period $2$ when $R$ is a local complete intersection.

EDIT: As Karl's comment indicated, perhaps a more realistic question is:

Question 2: Is the union of all minimal primes of $\text{Ext}^i_R(M,N)$ for $i>0$ a finite set?

One situation when the answer is trivially yes is when $\text{Spec}R$ has an isolated singularity.

I think the answer is no (even for question 2) in general, but not sure how to construct an example.

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    $\begingroup$ Long, can you say something about the "little bit more work" needed to show the c.i. case? Also, is the answer known if $R$ is c.i. (or hypersurface) on the punctured spectrum? $\endgroup$
    – Graham Leuschke
    Commented Jul 17, 2010 at 23:43
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    $\begingroup$ @Graham: It follows from a simple observation, see for example Cor 2.4 in math.ku.edu/~hdao/asymptotic.pdf and the fact that the total Ext module is Noetherian over the ring of (co)homology operators, defined in Eisenbud's paper. $\endgroup$
    – Hailong Dao
    Commented Jul 18, 2010 at 3:00
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    $\begingroup$ Long, is it clear that the union of the set of supports of these Ext modules is a finite set? $\endgroup$
    – Karl Schwede
    Commented Jul 18, 2010 at 4:11
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    $\begingroup$ Is minimal of $\cup \mathrm{Ass} \mathrm{Ext}^i_R(M, N)$ finite? $\endgroup$
    – Pham Hung Quy
    Commented Aug 12, 2012 at 17:15
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    $\begingroup$ This paper contains some relevant results which might be of interest doi.org/10.1017/S0305004115000778 ... apparently the question whether $\cup_{i\ge 0} Ass (Ext^i_R(M,R))$ is finite for a finitely generated $R$-module $M$ over a Noetherian local ring $R$, was first asked by Vasconcelos $\endgroup$
    – sdey
    Commented Jun 22, 2019 at 10:47

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