I have three questions about the Ext functor's properties.
(i) $Ext(H \oplus H',G) = Ext(H,G) \oplus Ext(H',G)$
(ii) $Ext(H,G) = 0$ if $H$ is free
(iii) $Ext(\mathbb{Z_n}, G) = G/nG$
There is a short proof in Hatcher page 195 and here are my questions:
(i) For (i), it is say this is done ("clearly") by the free resolutions
$0 \to F_2 \to F_1 \to F_0 \to H \to 0$
$\oplus$
$0 \to F_2' \to F_1' \to F_0' \to H' \to 0$
= $0 \to F_2 \oplus F_2' \to F_1 \oplus F_1' \to F_0 \oplus F_0' \to H\oplus H' \to 0$
Can someone write down the detail in this? I feel like we can't just write what I did, take the Ext and compare it to the Ext when we direct sum.
(ii) The proof is $0 \to H \to H \to 0$. I am guessing this is because the Ext functor is invariant under $H$, so the $F_2 \to F_1 \to F_0 \to \dots$ doesn't matter and we can take all of that as just $H$ since $H$ is free. Unrelated, but how does the Ext functor depend/invariant on $G$?
(iii)
The proof is the map which is a dual of $0 \to Z \to Z \to Z_n \to 0$
How do we get $Hom \to Ext$, the far left map? (Do not use vertical map)