Dualizing module and finiteness hypothesis

Question

Serre, in his Galois Cohomology, states:

Proposition 17. Let $n$ be an integer $\geq 0$. Assume:

(a) $\text{cd}(G) \leq n$

(b) For every $A \in C^f_G$, the group $H^n(G, A)$ is finite.

Then the functor $A \mapsto H^n(G, A)^*$ is representable on $C^f_G$ by an element $I$ of $C^t_G$.

This proposition is an easy consequence of the following Lemma:

Let $C$ be a noetherian abelian category, and let $T\colon C \to \text{Ab}$ be a contravariant additive left-exact functor. Then $T$ is representable by an object $I$ in $\varinjlim C$

I fail to see now how hypothesis (b) is relevant. It does not come into play on the proof of the lemma (which is stated with the wrong variance on the book), and only hypothesis (a) is necessary to apply it. It also appears on other references, such as Shatz's Proposition 65 of Profinite Groups, Arithmetic and Geometry.

I don't feel like the hypothesis is useless though, as Serre points out soon after that "if one had stuck to $p$-primary $G$-modules, one would have only needed the hypothesis $\text{cd}_p(G) \leq n$". But he could also be referring to hypothesis (a) only.

Answer 1

After banging my head over this for almost a month, I found an answer on Appendix I to Serre's book. It is an excerpt of a letter from Tate, and quoting it:

Tu es inutilement prudent en ce qui concerne le module dualisant: aucune hypothèse de finitude n'est necessaire.

You are unnecessarily cautious in your approach of the dualizing module: any finiteness hypothesis isn't necessary.

Thus, the hypothesis really isn't necessary. Also, in the English translation, there is the additional funny remark:

Let $T\colon C(R)^0 \to (\text{Ab})$ be an additive contrafunctor which transforms inductive limits into projective limits. Then $T$ is sexy, i.e. left exact, if and only if it is representable.