Let $A$ be a local noetherian commutative ring, let $x_1, \dots, x_r$ be elements of the maximal ideal of $A$, $I$ the ideal that they generate, $M$ a $A$-module of finite type such that $M/IM$ is of finite length.
The ideal $I$ defines a filtration of $A$ and $M$. Let $gr(A) = \bigoplus_{i\in\mathbb{N}} I^i / I^{i+1}$ and $gr(M) = \bigoplus_{i\in\mathbb{N}} I^iM / I^{i+1}M$ be the associated graded ring / module.
Let $\xi_1, \dots, \xi_r$ be the images of $x_1, \dots, x_r$ in $gr(A)_1 = I/I^2$. Then we can consider the Koszul complexes $K(\xi, gr(A))$ and $K(\xi, gr(M)) = K(\xi, gr(A)) \otimes gr(M)$.
In Local Algebra, p.58, (3.3), Jean-Pierre Serre write that the homology modules $H_p(\xi, gr(M))$ are of finite type over $gr(A)$. I do not understand why.
I agree that the modules $K_p(\xi, gr(M))$ are of finite type and that the result would follow from the fact that $gr(A)$ is noetherian, but I don't know if this last fact is true.
Thanks in advance