Exercise 1, § 10

Let $A$ be a ring, $J$ an ideal of $A$, $M$ an $A$-module. We denote by $H_{A,J}(M)$, or simply $H_J(M)$, the graded $A$-module $\lim_n \operatorname{Ext}_A(A/J^n, M)$. If $A$ is local noetherian, the graded $A$-module $H_{A,m_A}(M)$ is identified with $H_A(M)$.

a) Establish for $H_J(M)$ the properties analogous to those of $H_A(M)$ with respect to homomorphisms and exact sequences of $A$-modules (No. 1).

b) Define for every $i \geqslant 1$ isomorphisms $H_J^{i+1}(M) \to \lim_n \operatorname{Ext}_A^i(J^n, M)$.

c) Suppose the ring $A$ is local noetherian of dimension $d$, and the ideal $J$ generated by a completely secant sequence $(x_1, \ldots, x_r)$. Construct an isomorphism

which generalizes the isomorphism $\tau^i(M)$ of No. 2; in particular, one has $H_J^i(A) = 0$ for $i > r$.

d) For every ideal $K$ of $A$ contained in $J$, denote by $\rho_{K,J} : H_J(M) \to H_K(M)$ the homomorphism induced by the canonical mappings of $A/K^n$ into $A/J^n$. Let $I$ be an ideal of $A$; construct a long exact sequence of $A$-modules

where $\alpha(x) = (\rho_{I,I+J}(x), \rho_{J,I+J}(x))$ and $\beta(x, y) = \rho_{I \cap J, I}(x) - \rho_{I \cap J, J}(y)$.