Let $(R,m)$ be a noetherian local ring, and $M$ and $N$ be two finitely generated $R$-module. Then is it true that $\dim \text{Ext}^k(M,N)\leq \dim M-k$? If not does the reversed inequality hold?
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$\begingroup$ I don't get it: are you asking if the inequality holds for any $k$ or for some particular $k$ (and then which one?)? $\endgroup$– user26857Commented May 5, 2015 at 13:46
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It is not true.
Let $(R,m,k)$ be a Cohen-Macaulay local ring of dimension $d$ and $N$ be the canonical module of $R$. Let $M$ be a Cohen-Macaulay $R$-module of dimension $t$. Then, by Theorem 3.3.10 of Bruns-Herzog $\text{Ext}^{d-t}(M,N)$ is a Cohen-Macaulay $R$-module of dimension $t$.
So if $d-t\gt 0$, then $\dim \text{Ext}^{d-t}(M,N) =t =\dim M $
