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Is it true that $\text{Ext}(A,\Bbb Z)=\text{Ext}(A_{tor},\Bbb Z)$ for any abelian group $A$? Obviously this is true when $A$ is finitely generated, since $\text{Ext}(G,\Bbb Z)=0$ if $G$ is free. But I'm curious that whether or not this is true in the general case.

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Take $A = \mathbb Q$, since it is torsion free we have $\mathsf{Ext}(\mathbb Q_{\mathsf{tors}}, \mathbb Z) = 0$, but $\mathsf{Ext}(\mathbb Q,\mathbb Z) \neq 0$. See here.

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