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Let R be a noetherian ring and M , N be finitely generated $R$-modules. Then what is the relation between $Ass Ext^i_R(M,N)$ and $Ass M, Ass N$?

$Ass$ means set of associated prime ideals. It's well known $Ass Hom_R(M,N) \subseteq Supp M \cap Ass N $.

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It would be most helpful if you give some motivation and/or background. The way it phrased now makes it look like you want us to do your homework... –  Vladimir Dotsenko Feb 28 '12 at 15:06
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You would really think about me. –  Stella Feb 28 '12 at 20:56
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This is probably not a homework question. –  Hailong Dao Feb 29 '12 at 3:09
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Actually there's something stronger true than what you say in the last line. Namely, if $R$ is any commutative ring, $M$ is a finitely presented module, and $N$ is any module, then $Ass Hom_R(M,N) = Supp M \cap Ass N$. This is indeed well-known. For instance, it appears as an exercise in the book by Bruns and Herzog. –  Neil Epstein Mar 3 '12 at 21:22
    
Please, I want to know the proof of the fact AssHomR(M,N)=SuppM∩AssN if M,N are finitely generated R modules given R is noetherian. –  user40948 Oct 6 '13 at 13:50
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up vote 3 down vote accepted

The strict answer to your question is no in general. Take a very special case, $R=k[x,y]$, $M=m$ some maximal ideal of $R$, $N=R$. Then $Ass(M) = Ass(N) = \{(0)\}$, but it is not hard to see $Ext^1(M,N) \cong R/m$, so $Ass(Ext^1(M,N)) = \{m\}$.

However, the general question of understanding the associated primes of Ext is harder and I happen to think about it recently. When the ring is regular, one can get a complete (but complicated) description of the support of $Ext^i(M,N)$ based only on the depth of the modules $M,N$ locally at the primes in $Spec(R)$. This was announced by Auslander at the end of his ICM 1962 speech. Sadly enough, the paper he referred to seems to be mysteriously lost.

Shameless plug: Together with Ryo Takahashi, we accidentallly managed to recover Auslander's Theorem 3 from the speech cited above (which is about support of Tor). I am optimistic that his final paragraph can be deciphered in the near future.

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