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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 $i=1,\cdots n-1$?

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Maybe I'm misunderstanding what the process is, but it seems you took a projective resolution of $k[x_1, \ldots, x_n]$, then tensored that with $V$, then took homology. Isn't this just computing $Tor^{k[x_1, \ldots, x_n]}_i(V, k)$? If it is, then just make a torsion module over $k[x]$ and $H_0(K\otimes V)$ will be $V$ and hence non-zero. – Matt Aug 5 '12 at 17:23
sorry Matt. question error. you corrent. $H_0(K\otimes V)$ be nonzero. – Sang Cheol Lee Aug 6 '12 at 1:47

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