Proposition 5

Let $\rho : A \to B$ be a local homomorphism of noetherian local rings, such that the residual extension $\kappa_A \to \kappa_B$ induced by $\rho$ is of finite degree. Let $I_A$ be a Matlis $A$-module.

a) Let $I_B$ denote the sub-$B$-module of $\mathrm{Hom}_A(B, I_A)$ formed by the $A$-homomorphisms from $B$ into $I_A$ whose kernel contains a power of $m_B$. Then $I_B$ is a Matlis $B$-module.

b) Let $M$ be a $B$-module. The canonical mapping

defined by $\alpha(u)(m) = u(m)(1)$ induces a $B$-isomorphism of $D_B(M) = \mathrm{Hom}_B(M, I_B)$ onto the sub-$B$-module $\mathrm{Hom}_A^{cont}(M, I_A)$ of $D_A(M) = \mathrm{Hom}_A(M, I_A)$ consisting of mappings $f : M \to I_A$ such that for every element $m$ of $M$, there exists an integer $n \geqslant 0$ such that $f(m^n_B m) = 0$.

The above condition on $f$ means that $f$ is continuous when $I_A$ is given the discrete topology and $M$ the finest topology inducing on each finitely generated submodule the $m_B$-adic topology, which justifies the notation. Analogously, the condition $g \in I_B$ means that $g$ is continuous when $I_A$ is given the discrete topology and $B$ the $m_B$-adic topology.

Let us prove a). For every $B$-module of finite length $M$, denote by $T(M)$ the $B$-module $\mathrm{Hom}_A(M, I_A)$; for every $B$-linear mapping $f : M \to N$ between $B$-modules of finite length, denote by $T(f) : T(N) \to T(M)$ the $B$-linear mapping $\mathrm{Hom}_A(f, 1_{I_A})$. The verification of conditions FD 1) to FD 4) of No. 5 is immediate. Moreover, for every $B$-module of finite length $N$, one has $\mathrm{long}_A(N_{[A]}) = \mathrm{long}_B(N) \ [\kappa_B : \kappa_A]$; since one has $\mathrm{long}_A(T(M)) = \mathrm{long}_A(M)$ (No. 3, th. 2), one deduces that $\mathrm{long}_B(T(M)) = \mathrm{long}_B(M)$, which implies FD 5). We may therefore apply theorem 3 of No. 5; one has

so that the Matlis $B$-module $\varprojlim T(B/m_B^n)$ is identified with the sub-$B$-module $I_B$ of $\mathrm{Hom}_A(B, I_A)$, which proves a).

Let us prove b). The mapping $\alpha$ is the inverse of the canonical isomorphism

which associates to $v \in \mathrm{Hom}_A(M, I_A)$ the map $v'$ of $M$ into $\mathrm{Hom}_A(B, I_A)$ such that $v'(m)(b) = v(bm)$ (A, II, p. 74, prop. 1). In order that $v'$ take its values in $I_B$, it is necessary and sufficient that $v$ belong to $\mathrm{Hom}_A^{cont}(M, I_A)$, whence b).