Example 4

If $A$ is a Gorenstein ring, for example a regular ring, one can apply Example 3 by taking $\Omega = A$ and $d = \dim(A)$ (§ 3, No. 7, Prop. 11).

For every integer $n \geqslant 0$, set $I_n = T(A/\mathfrak{m}_A^n)$. For $m \geqslant n$, denote by $p_{mn} : A/\mathfrak{m}_A^m \longrightarrow A/\mathfrak{m}_A^n$ the canonical surjection and by $i_{mn} : T(A/\mathfrak{m}_A^n) \longrightarrow T(A/\mathfrak{m}_A^m)$ the $A$-linear mapping $T(p_{mn})$. It is injective by FD 4), and one has $i_{mn} \circ i_{np} = i_{mp}$ for $m \geqslant n \geqslant p$ by FD 3). Let $I = \varprojlim T(A/\mathfrak{m}_A^n)$ be the inductive limit $A$-module of the system $((I_n), (i_{mn}))$. For $n \geqslant 0$, the canonical mapping $I_n \to I$ is injective; we identify $I_n$ with its image in $I$, so that $I$ is the increasing union of the $I_n$.

Let M be an A-module of finite length, and n an integer $\geqslant 0$ such that $m_A^n M = 0$. For $x \in M$, let us denote by $\varphi_{M,x}^n$ the a-linear mapping of $A/m_A^n$ into M which maps the class of 1 to x. The mapping $T(\varphi_{M,x}^n) : T(M) \to I_n$ is a-linear, and one has $T(\varphi_{M,a x}^n) = a T(\varphi_{M,x}^n)$ for $a \in A$ by FD 1). Hence the mapping $(x, u) \mapsto T(\varphi_{M,x}^n)(u)$ of $M \times T(M)$ into I is A-bilinear. It does not depend on the choice of the integer n: in fact, for every integer $q \geqslant n$ and every element x of M, one has $\varphi_{M,x}^q = \varphi_{M,x}^n \circ p_{q n}$, whence by FD 3) $T(\varphi_{M,x}^q) = i_{q n} \circ T(\varphi_{M,x}^n)$. We deduce an a-linear mapping

satisfying $\theta_M(u)(x) = T(\varphi_{M,x})(u)$ for $u \in T(M),\ x \in M$.