For a graded finitely generated $k[x_1, \cdots, x_n]$ module $V$, I know that $ b_{i,p}(V)=\operatorname{dim}_k H_i(K\otimes V)_p$ where $K$ be the Koszul complex of $k$.
I also know that $K$ is minimal. then is always $H_i(K\otimes V)=0$ for $i=1,\cdots n-1$?