# finiteness of Koszul groups

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?

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.