Proposition 3

Let $A$ be a noetherian ring, $p$ a prime ideal of $A$, and $e : A/p \to I$ an injective envelope of the $A$-module $A/p$. For every integer $n \geq 0$, let $I_n$ denote the submodule of $I$ consisting of the elements annihilated by $p^n$.

a) The $A$-module $I$ is the union of the $I_n$. The injection $A / \mathfrak{p} \to I_1$ extends to an isomorphism of $\kappa(\mathfrak{p})$ onto $I_1$; let us identify $\kappa(\mathfrak{p})$ with $I_1$ by means of this isomorphism. For each integer $n \geqslant 0$, the structure of $A / \mathfrak{p}$-module of $I_{n+1}/I_n$ arises by restriction of scalars from a unique structure of $\kappa(\mathfrak{p})$-vector space; the canonical homomorphism $\beta_{I,-n}: I_{n+1}/I_n \longrightarrow \mathrm{Hom}_{A/\mathfrak{p}}(\mathfrak{p}^n/\mathfrak{p}^{n+1}, \kappa(\mathfrak{p}))$ is an isomorphism of finite-dimensional $\kappa(\mathfrak{p})$-vector spaces.

b) There exists a unique structure of $\widehat{A_p}$-module on $I$ inducing its structure of $A$-module. The canonical homomorphism $\widehat{A_p} \longrightarrow \mathrm{End}_A(I)$ is bijective.

According to A, X, p. 20, Example 1, the $A/\mathfrak{p}$-module $\kappa(\mathfrak{p})$ is an injective envelope of $A/\mathfrak{p}$. It therefore follows from Prop. 2 that $I_1$ identifies with $\kappa(\mathfrak{p})$, that $I$ is the union of the $I_m$, and that for each integer $n \geqslant 0$, $\beta_{I,-n}$ is an isomorphism of $A/\mathfrak{p}$-modules. For every nonzero element $a$ of $A/\mathfrak{p}$, the homothety of ratio $a$ is invertible in $\mathrm{Hom}_{A/\mathfrak{p}}(\mathfrak{p}^n/\mathfrak{p}^{n+1}, \kappa(\mathfrak{p}))$, hence also in $I_{n+1}/I_n$, which completes the proving of a).

Let $s \in A - \mathfrak{p}$. Since the homothety $s_{A/\mathfrak{p}}$ is injective, the trace of $\mathrm{Ker}\, s_I$ on $A/\mathfrak{p}$ is zero, whence it follows that the homothety $s_I$ is injective. Then $s_I$ is a direct factor submodule of $I$ (A, X, p. 19, Cor. 4), hence equal to $I$ since $I$ is indecomposable (No. 1, Prop. 1), so that the homothety $s_I$ is bijective. There therefore exists a unique $A_p$-module structure on $I$ inducing its $A$-module structure; it extends uniquely to a $\widehat{A_p}$-module structure (Lemma 2).

For each integer $n$, one deduces from the canonical ring homomorphism $A_p \longrightarrow \mathrm{End}_A(I)$ an $A$-linear mapping $\alpha_n : A_p/\mathfrak{p}^n A_p \longrightarrow \mathrm{Hom}_A(I_n, I)$. Consider the commutative diagram with exact rows

where $\alpha'_{n+1}$ is the homomorphism induced by $\alpha_{n+1}$. Consider the canonical $\kappa(\mathfrak{p})$-bilinear mapping

(formula (1)). The linear mapping $I_{n+1}/I_n \longrightarrow \mathrm{Hom}_{\kappa(\mathfrak{p})}(\mathfrak{p}^n A_p/\mathfrak{p}^{n+1} A_p, I_1)$ associated with it on the left identifies with $\beta_{I,-n}$, and that associated with it on the right is $\alpha'_{n+1}$. Since $\beta_{I,-n}$ is bijective by a), the same is true of $\alpha'_{n+1}$; one then deduces from the above diagram, by induction on $n$, that $\alpha_n$ is an isomorphism for every $n$. Since $I$ is the union of the $I_n$, the canonical mapping $\mathrm{End}_A(I) \to \varprojlim \mathrm{Hom}_A(I_n, I)$ is bijective; the ring homomorphism $\widehat{A_p} \to \mathrm{End}_A(I)$, which identifies with the projective limit of the mappings $\alpha_n$, is therefore bijective.