Exercise 11, § 10

Soient $A$ a local noetherian ring, $I$ a Matlis $A$-module, $M$ a finitely generated $A$-module. Put $\mu^p = \dim_{\kappa_A} \mathrm{Ext}_A^p(\kappa_A, M)$ for all $p \geqslant 0$ ($cf.$ § 8, Exercise 4). Prove that the graded $A$-module $H_A(M)$ is identified with the homology of a complex $\ldots \to 0 \to I^{\mu^0} \to I^{\mu^1} \to \ldots$ (\emph{loc. cit.}). In particular, if $M$ is nonzero, one has $\mu^p \neq 0$ for $p = \mathrm{prof}(M)$ and $p = \dim(M)$ (Exercise 4).

11) Let $A$ be a local noetherian ring, $M$ a finitely generated $A$-module, of dimension $d$; suppose that $M$ is a Buchsbaum module ($§ 2$, Exercise 7).

a) Prove that $H^i_A(M)$ is annihilated by $m_A$ for $i \neq d$ (first treat the case $d = 1$, then reason by induction on $d$).

b) Let $x$ be a secant element for $m$; define for $0 \leqslant i \leqslant d-2$ exact sequences $0 \to H^i_A(M) \to H^i_A(M/xM) \to H^{i+1}_A(M) \to 0$.

c) Prove the formula $i(M) = \sum_{i=0}^{d-1} \binom{d}{i} \lg(H^i_A(M))$ (reason by induction on the integer $d \geqslant 1$, using b) and Exercise 8, d) of the $§ 2$).

1 We do not know whether every local noetherian ring satisfies (CM).