Let $(R,\mathfrak m)$ be a Commutative Noetherian local ring. Let $K^R$ be the Koszul complex on a minimal generating set of $\mathfrak m$. Then, is it true that $H_i(K^R)=0$ for all $i>\mu(\mathfrak m)-\text{depth } R $ ?
This is clearly true if $\text{depth } R=0$ as the last non-zero module in the Koszul complex is the $\mu(\mathfrak m)$-th place. But I'm not sure what happens if $R$ has positive depth.
Please help.