Example 3

Let $\Omega$ be an $A$-module and $d$ an integer $\geqslant 0$ such that $\operatorname{Ext}_A^i(\kappa_A, \Omega)$ is zero for $i \neq d$ and has length 1 for $i = d$. Set $T(M) = \operatorname{Ext}_A^d(M, \Omega)$ for every $A$-module $M$ of finite length and $T(f) = \operatorname{Ext}_A^d(f, 1_\Omega)$ for every $A$-linear mapping $f$ between $A$-modules of finite length. Then $\operatorname{Ext}_A^i(M, \Omega) = 0$ for every $A$-module $M$ of finite length and every $i \neq d$, and conditions FD 1) to FD 5) are satisfied: it is enough, in fact, to apply the preceding example to the case where $C$ is the canonical injective resolution of $\Omega$.