Proposition 6

Suppose $H_0$ local and artinian, $H$ finitely generated as an $H_0$-algebra and $M$ finitely generated as an $H$-module.

a) Let $a_1, ..., a_n$ be elements of $H$, homogeneous of degrees $> 0$, and let $\varphi$ be the homomorphism (of $H_0$-algebras) of $H_0[X_1, ..., X_n]$ into $H$ which maps $X_i$ to $a_i$ for $1 \leq i \leq n$. The $S^{-1}H$-module $S^{-1}M / \sum_{i=1}^n (a_i/1).S^{-1}M$ is of finite length if and only if $\varphi_*(M)$ is a finitely generated module over $H_0[X_1, ..., X_n]$.

b) There exists a family $(a_1, ..., a_d)$ of elements of $H$, all homogeneous of the same degree $> 0$, with $d = \dim_H(M)$, and such that $(a_1/1, ..., a_d/1)$ is a maximal secant sequence for the $S^{-1}H$-module $S^{-1}M$. If moreover $H$ is generated by $H_1$ as an $H_0$-algebra, and if the residue field of $H_0$ is infinite, one can take the $a_i$ of degree 1.

a) Put $N = M / \sum_{i=1}^n a_i M$. We have $\dim_H(N) = \dim_{S^{-1}H}(S^{-1}N)$ by prop. 4. Consequently, the $S^{-1}H$-module $S^{-1}N$ is of finite length if and only if the $H$-module $N$ is of finite length, that is to say if and only if $N$ is a finitely generated module over $H_0$. If $\varphi_*(M)$ is the module over $H_0[X_1, ..., X_n]$ deduced from $M$ by the homomorphism $\varphi : H_0[X_1, ..., X_n] \to M$, we have $N = \varphi_*(M)/\sum_{i=1}^n X_i \cdot \varphi_*(M)$. Consequently (A, II, p. 171, cor. 3 and remark) $\varphi_*(M)$ is a finitely generated module over $H_0[X_1, ..., X_n]$ if and only if $N$ is a finitely generated module over $H_0$. This proves a).

To prove b), we shall first establish a lemma.