Lemma 3

Let I be a Matlis A-module. Then:

a) I is a Matlis $\widehat{A}$-module; b) the A-module I is artinian and a cogenerator (A, X, p. 18, def. 3).

Since the A-module I is injective, the A/m_A^n-module I_n is injective for each n (No. 2, lemma 1, a)). As I_n is the set of elements of I annihilated by m_A^n, the $\widehat{A}$-module I is injective (lemma 1, b)). It is indecomposable over $\widehat{A}$ since it is so over A; as it contains the sub-$\widehat{A}$-module I_1 isomorphic to $\kappa_A$, we have $m_{\widehat{A}} \in \operatorname{Ass}_{\widehat{A}}(I)$, hence $\operatorname{Ass}_{\widehat{A}}(I) = \{ m_{\widehat{A}} \}$ (prop. 1), whence a).

Let us prove now that I is artinian. To every sub-A-module M of I, let us associate the graded ideal $a_M$ of gr(A) defined as follows: an element of gr(A)_n belongs to $(\mathfrak{a}_M)_n$ if it is annihilated by all the linear forms $\beta(x)$, where $x$ runs through $((M \cap I_{n+1}) + I_n)/I_n$. Let $M$ and $N$ be submodules of $I$ such that $N \subset M$; then $\mathfrak{a}_M \subset \mathfrak{a}_N$. Suppose $\mathfrak{a}_M = \mathfrak{a}_N$; we have $(M \cap I_{n+1}) + I_n = (N \cap I_{n+1}) + I_n$ for every $n$ since $\beta$ is an isomorphism. By induction on $n$ we deduce from this that $M \cap I_{n+1} = N \cap I_{n+1}$ for every $n$, whence finally $M = N$.

This being so, let $M_0 \supset M_1 \supset \ldots \supset M_n \supset \ldots$ be a decreasing sequence of sub-A-modules of $I$; the increasing sequence $\mathfrak{a}_{M_0} \subset \mathfrak{a}_{M_1} \subset \ldots$ is stationary, since $\mathrm{gr}(A)$ is a finitely generated $\kappa_A$-algebra. The sequence $(M_i)_{i \geq 0}$ is therefore stationary, which implies that the A-module $I$ is artinian. Finally, the A-module $I$ is a cogenerator by virtue of A, X, p. 18, prop. 12.

Let $M$ be an A-module. Recall (A, VIII, § 4, No. 6) that the socle of $M$ is the sum of the simple submodules of $M$, that is to say the set of elements of $M$ annihilated by $m_A$; it is a $\kappa_A$-vector space, canonically isomorphic to $\mathrm{Hom}_A(\kappa_A, M)$.