Exercise 9, § 2

Let $A$ be a local noetherian ring.

a) Let $M$ be a Macaulay finitely generated $A$-module, $N$ a submodule of $M$ containing $m_A M$. Prove that $N$ is a Buchsbaum module, and that one has $i(N) = (\dim(M) - 1)[M/N : \kappa_A]$.

b) If $A$ is a Macaulay ring, the ideal $m_A$ is a Buchsbaum module, which is not Macaulay if $\dim(A) \geqslant 2$.

c) Let $k$ be a field; take for $A$ the subring of $k[[X, Y]]$ generated by $X^4, X^3Y, XY^3, Y^4$. One has $\dim(A) = 2$, $\operatorname{prof}(A) = 1$, and $A$ is a Buchsbaum ring, with $i(A) = 1$ (apply a) taking for $M$ the integral closure of $A$).