Exercise 8, § 10

On retains the notations of the preceding exercise, assuming in addition that the ring $A$ is noetherian. We denote by $x$ the ideal generated by $x$.

a) Let $m$ be an integer. Prove that there exists an integer $n_0 \geq m$ such that for $n \geq n_0$, the morphism $\Lambda(\Delta(x)^{n-m}) : K_\bullet(x^n, A) \longrightarrow K_\bullet(x^m, A)$ induces $0$ on the homology of degree $\geq 1$ (observe that one has $\Lambda^p(\Delta(x)^{n-m})(K_p(x^n, A)) \subset x^{n-m} K_p(x^m, A)$; deduce from the Artin-Rees lemma that there exists an integer $n_0$ such that $Z_p(x^m, A) \cap x^{n_0} K_\bullet(x^m, A)$ is contained in $x^{rm} Z_p(x^m, A)$, and consequently in $B_p(x^m, A)$).

b) Define a canonical isomorphism of $\varprojlim_n \operatorname{Ext}_A^p(A/x^n, M)$ onto $H^p(x^\infty, M)$ (reduce to proving that $H^p(x^\infty, M)$ is zero for $p \geq 1$ and $M$ injective; in this case observe that $H^p(x^\infty, M)$ identifies with the inductive limit of the $\operatorname{Hom}_A(H_p(x^n, M))$, and use a)). In particular, if $A$ is local and if $x$ is a defining ideal of $A$, the graded $A$-module $H_A(M)$ is canonically isomorphic to $H^\bullet(x^\infty, M)$.

c) Suppose the ring $A$ is local noetherian, of dimension $d$; let $x = (x_1, \ldots, x_d)$ be a maximal sequence of elements of $\mathfrak{m}_A$. Prove that the $A$-module $H^d_A(M)$ identifies with the limit of the inductive system of $A$-modules $(M_p, u_{qp})$, where $M_p$ is the quotient module $M/(x_1^p M + \ldots + x_d^p M)$ and $u_{qp} : M_p \to M_q$ ($q \geq p$) the homomorphism induced by the homothety of ratio $(x_1 \ldots x_d)^{q-p}$.