Remark 1

Let $X$ be the topological space $\mathrm{Spec}(A)$, $\mathscr{O}_X$ the structural sheaf of rings and $\widetilde{M}$ the $\mathscr{O}_X$-module associated with $M$. The graded $A$-module $H_A(M)$ is identified with the module $H_{\{m_A\}}(X, \widetilde{M})$ of cohomology with support in the closed point $m_A$ of $X$.

For every homomorphism $f : M \to N$ of $A$-modules, the mappings $\mathrm{Ext}_A(1_{A/a}, f) : \mathrm{Ext}_A(A/a, M) \longrightarrow \mathrm{Ext}_A(A/a, N)$ form an inductive system of graded linear mappings. By passing to the inductive limit, one obtains a graded homomorphism $H_A(f) : H_A(M) \to H_A(N)$. For every sequence $M \xrightarrow{f} N \xrightarrow{g} P$ of $A$-modules and homomorphisms, one has $H_A(g \circ f) = H_A(g) \circ H_A(f)$. Let

be an exact sequence of $A$-modules. By $A$, X, p. 90, Prop. 8, the connecting homomorphisms of the extension modules $\mathrm{Ext}_A(A/a, P) \longrightarrow \mathrm{Ext}_A(A/a, M)$ form an inductive system of $A$-linear mappings, graded of (ascending) degree +1. By passing to the inductive limit, one deduces an $A$-homomorphism $\partial(\mathscr{E}) : H_A(P) \to H_A(M)$, graded of degree +1, which makes exact the sequence of homomorphisms

Let M be an a-module. For every ideal $\alpha$ of A, the A-module $\mathrm{Hom}_A(A/\alpha, M)$ is canonically identified with the submodule of M consisting of the elements annihilated by $\alpha$. Thus $H_A^0(M)$ is identified with the submodule of M consisting of the elements $m$ which are annihilated by a power of $\mathfrak{m}_A$, that is to say such that $\mathrm{long}_A(Am) < +\infty$. In particular one has $H_A^0(M) = M$ when M is artinian.