Exercise 4, § 3

Let $A$ be a noetherian ring, $M$ a nonzero $A$-module of finite type. One has $\mathrm{grade}(M) \leq \mathrm{dp}_A(M)$; one says that $M$ is perfect if there is equality. a) Let $d$ be an integer; prove that the following conditions are equivalent: (i) the $A$-module $M$ is perfect of projective dimension $d$; (ii) one has $\mathrm{Ext}_A^p(M, A) = 0$ for $p \neq d$; (iii) for every prime ideal (resp. maximal ideal) $p$ of the support of $M$, the $A_p$-module $M_p$ is perfect of projective dimension $d$. (Prove first the equivalence of (i) and (ii).) b) Let $J$ be an ideal of $A$ generated by a completely secant sequence, $m$ an integer $\geq 1$. Prove that the $A$-module $A/J^m$ is perfect. c) If $M$ is perfect, its associated prime ideals are the ideals $p$ such that $\mathrm{prof}(A_p) = \mathrm{dp}_A(M)$. d) For every perfect module $N$ of projective dimension $d$, put $N' = \mathrm{Ext}_A^d(N, A)$. Prove that $N'$ is perfect, of projective dimension $d$, and that $(N')'$ is isomorphic to $N$ (consider a projective resolution of $N$). One has $\mathrm{Ass}(N') = \mathrm{Ass}(N)$. e) Let $x$ be a cancellable element of the radical of $A$, such that the homothety $x_M$ is injective. In order that the $A$-module $M$ be perfect of projective dimension $d$, it is necessary and sufficient that the same be true of the $(A/xA)$-module $M/xM$. f) A Macaulay module of finite projective dimension is perfect (reduce to the local case, and reason by induction on $\dim(M)$). g) If $A$ is a Macaulay ring, the perfect modules are the Macaulay modules of finite projective dimension.