Exercise 7, § 2

Let $A$ be a noetherian local ring, $M$ a finitely generated a-module. We say that a sequence $(x_1, \ldots, x_n)$ is weakly regular for $M$ if, for $i = 1, \ldots, n$, the kernel of the homothety of ratio $x_{i+1}$ in $M/(x_1 M + \ldots + x_i M)$ is annihilated by $m_A$.

a) If $n \leqslant \dim(M)$, prove that a weakly regular sequence $(x_1, \ldots, x_n)$ for $M$ is sécant for $M$ (reason by induction on $n$). Give an example of a weakly regular non-sécant sequence (one may take $A = M = k[X]/(X^2)$, where $k$ is a field).

b) We say that $M$ is a Buchsbaum module if every sécant sequence for $M$ is weakly regular for $M$. A Cohen-Macaulay module is a Buchsbaum module; if $M$ is a Buchsbaum module, the $A_p$-module $M_p$ is Cohen-Macaulay for every prime ideal $p \neq m_A$.

c) Let $M$ be a Buchsbaum module, $(x_1, \ldots, x_n)$ a sécant sequence for $M$; prove that $M/(x_1 M + \ldots + x_n M)$ is a Buchsbaum module.

d) For $M$ to be a Buchsbaum module, it is necessary and sufficient that, for every maximal sécant sequence $(x_1, \ldots, x_n)$ for $M$, one has in $M_n = M/(x_1 M + \ldots + x_{n-1} M)$ the equality $\mathrm{Ker}(x_n)_{M_n} = \mathrm{Ker}(x_n^2)_{M_n}$ (first treat the case $n = 1$, observing that $\mathrm{Ker}(x_1)_M$ is then of finite length and reasoning by induction on this length. In the general case, complete every sécant sequence $(x_1, \ldots, x_i)$ to a maximal sécant sequence $(x_1, \ldots, x_n)$, and consider the sequences $(x_1, \ldots, x_{i-1}, x_{i+1}^k, \ldots, x_n^k, x_i)$ for $k \geqslant 1$.