Example 2

Let $C$ be an injective complex of $A$-modules and $d$ an integer such that $H^i(\operatorname{Homgr}_A(\kappa_A, C))$ is zero for $i \neq d$ and is of length 1 for $i = d$. For every $A$-module $M$ of finite length, one has $H^i(\operatorname{Homgr}_A(M, C)) = 0$ for $i \neq d$; let us argue in fact by induction on the length of $M$, supposed $> 0$; there exists an exact sequence of $A$-modules $0 \to \kappa_A \to M \to N \to 0$, which gives rise to an exact sequence of complexes

and the conclusion follows from the induction hypothesis applied to $N$.

Let $T(M) = H^d(\operatorname{Homgr}_A(M, C))$ for every $A$-module $M$ of finite length, and $T(f) = H^d(\operatorname{Homgr}_A(f, 1_C))$ for every $A$-linear mapping $f$ between $A$-modules of finite length; conditions FD 1) to FD 5) are satisfied.