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I am reading journal of number theory and the paper quites a result.

$$ \text{Ext}^1(\mathbb{Q}, \mathbb{Z}) = \hat{\mathbb{Z}}/\mathbb{Z} = \mathbb{A}_f / \mathbb{Q} $$

Unfortunately, before I prove this result, this statement contains a number of concepts. What is is the Ext group measuring in this context? How do we construct elements of $\hat{\mathbb{Z}}$ which is the profinite completion of $\mathbb{Z}$.

The paper also mentions a group action of $\text{PSL}_2(\mathbb{Q})$ on $\mathbb{A}_f/ \mathbb{Q}$ which looks like the quotient of the finite adeles by rationals under multiplication: $$ z \cdot \left( \begin{array}{cc} a & b \\ c & d\end{array} \right) = \frac{b + dz}{a + cz} \in \mathbb{A}_f / \mathbb{Q} $$

There is a note (and blog) where they walk us though the proof of $(*)$. I still have a lot of questions. Here are a random few observations from it:

  • $\mathbb{Q} \otimes \mathbb{Q} \simeq \mathbb{Q}$

  • $\mathbb{Z}/n \otimes \mathbb{Q} = 0$

  • $G \otimes H \simeq H \otimes G$ this seems trivially true.

  • $\mathbb{Q}/\mathbb{Z} \simeq \bigoplus_p \mathbb{Z}/p^\infty$ where $\mathbb{Z}/p^\infty := \mathbb{Z}[p^{-1}]/\mathbb{Z}$

There are some discussion of exact sequences:

  • $\text{Hom}(\mathbb{Q}, - )$ is a functor.
  • $ 0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0 $ this exact sequence (of additive groups, $(\mathbb{Z},+)$ and $(\mathbb{Q},+)$) seems.

That these objects exists at all is quite interesting. Looks on the surface quite elementary (e.g. unique factorization).

Maybe related:

cactus314
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    "What does it measure in this context?" : what context, number theory in general ? – Maxime Ramzi Aug 22 '19 at 16:09
  • In general that Ext measures how many maps (group homomorphisms) $\mathbb{Z}\rightarrow\mathbb{Z}$ comes from maps $\mathbb{Q}\rightarrow\mathbb{Z}$. I don't know what else to say besides it. – Rafael Aug 22 '19 at 18:28
  • $\hat{\mathbb{Z}}$ is the set of limits of sequences of integers that converge modulo every $n$, a ring with the termwise addition and multiplication of those sequences. How do you define $\text{Ext}^1(\mathbb{Q}, \mathbb{Z})$ concretely – reuns Aug 23 '19 at 09:25

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