Suppose $A=k[x_1,..,x_n]_{(x_1,..,x_n)}$, it is a regular local ring of dimension $n$. Let $B=A/I$ be a quotient ring of Krull dimension $r$. How to show $\operatorname{Ext}_A^i(B,A)=0$ for $i<n-r$?
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We have $\dim A=\dim A/I+\operatorname{ht}I$, so $\operatorname{ht}I=n-r$. Then $\operatorname{grade}I=n-r$ and now use the Rees theorem (see Bruns and Herzog, Theorem 1.2.5).
user26857
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It's enough to assume that $A$ is local and Cohen-Macaulay. – user26857 Aug 23 '15 at 18:15