A basic question about Koszul homology from Matsumura's Commutative Ring Theory

In Theorem 16.5(ii) it is assumed that $(A,m)$ is a local ring and $x_1,\ldots,x_n \in m$, and $M$ is a finite $A$-module. Then it is claimed without much explanation that the Koszul homology groups $H_p(X,M)$ are finite $A$-modules for all $p$. Why is this so obviously true?


1 Answer 1


If $A$ is Noetherian, then this is indeed obvious: the homology group $H_p(X,M)$ is defined as $\ker g/\operatorname{im}f$ for certain maps $$M^i\stackrel{f}\to M^j\stackrel{g}\to M^k$$ and certain $i,j,k\in\mathbb{N}$. Since $M$ is finitely generated, so is $M^j$, and thus so is $\ker g$ since $A$ is Noetherian, and thus so is $H_p(X,M)$.

If $A$ is not assumed to be Noetherian, then this is not true. For instance, $A$ could be $k[t_1,t_2,t_3,\dots]/(t_1,t_2,t_3,\dots)^2$ for a field $k$. Then for $M=A$, $n=1$, and $x_1=t_1$, the Koszul complex is just $$0\to A\stackrel{t_1}\to A\to 0.$$ So $H_1(X,A)$ is just the kernel of $t_1:A\to A$ which is the maximal ideal $(t_1,t_2,t_3,\dots)$, which is not finitely generated.


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