Ext of the rationals Q is a vector space over the rationals

Question

How to see that Ext$_Z^1$(Q, A) is a vector space over Q (where Q is the rationals) for any abelian group A? Any help would be appreciated!

Answer 1

If $a\in\Bbb Q$ then $a$ induces a map $\mu_a$ "multiplication by $a$" from $\Bbb Q$ to $\Bbb Q$. As $\text{Ext}$ is a bifunctor, this together with the identity map on $A$ induces a map $\mu_a^*:\text{Ext}^1(\Bbb Q,A)\to \text{Ext}^1(\Bbb Q,A)$. We define scalar multiplication $a\cdot\omega=\mu_a^*(\omega)$ for $\omega\in\text{Ext}^1(\Bbb Q,A)$. This makes $\text{Ext}^1(\Bbb Q,A)$ into a vector space over $\Bbb Q$.