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I've wanted to understand the concept of the Dualizing module in the theory of Galois Cohomology. There are many references on it and of them all Neukirch's Cohomology of Number Fields seems to be elaborate but it seems to be most involved.

So instead I want to learn it from the Serre's Galois Cohomology at first. But I can't understand all the arguments of the sketch of proof there.

Let me elaborate what I have understood, and what I have not. It's easy to understand that the existence of the Dualizing Module follows from the statement saying that If $C$ is a noetherian abelian category, and If $T : C \rightarrow (Ab)$ is a contravariant right-exact functor from $C$ to $(Ab)$. Then the functor $T$ is representable by an object that is a direct limit of objects in $C$.

  1. The definition of minimal pair, and the ordering between them. And the uniqueness of the map between ordered pair of objects of $C$ makes sense to me. If there are 2 different maps $u_1, u_2$ from $A$ to $A'$ with $T(u_1)(x')=T(u_2)(x')=x$, then that means $x'$ comes from the subgroup of kernel of $T(u_1-u_2)$ which is isomorphic to $T(coker(u_1-u_2))$(which is not the same as $T(A')$) contradicting the minimality of $(A',x')$

  2. I guess I have understood the statement that the set of minimal pairs is a filtered ordered set. Given $(A_1,x_1)$, $(A_2, x_2)$ two minimal pairs. We choose $A_3=A_1 \oplus A_2$, then $T(A_3)=T(A_1)\oplus T(A_2)$ and we can let $x_3=(x_1,x_2)$. Although Serre said that the Noetherian condition of $C$ is used in the later part, I guess that it is also useful here to guarantee that for $(A_3, x_3)$, it has some minimal $(A_4, x_4)$ with a epimorphism $\pi$ from $A_3$ to $A_4$ with $T(\pi)(x_4)=x_3$. Because for a Neotherian category the sequence of epimorphisms always are stationary.

After that, I can't follow the sketch of proof.

I hope that someone can help me about this. Thank you.

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  • $\begingroup$ This is in I.§3.5. Do you understand the construction of the dualising object and that it represents $T$? According to the book, the proof can be found in Gabriel's thesis Des catégories abéliennes, Chapter II.§4 numdam.org/item/BSMF_1962__90__323_0. $\endgroup$
    – user19475
    Commented Dec 5, 2018 at 8:43
  • $\begingroup$ @TKe yes, but I hoped that there might be someone who are already familiar with the argument. Maybe I have to start reading the paper. Thank you $\endgroup$
    – gualterio
    Commented Dec 5, 2018 at 13:10

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