Determine $\operatorname{Ext}_{\mathbb Z}^i(A,B)$ using projective resolutions

Question

I know that there are many resources (such as Chap. 17.1 Dummit & Foote, in particular pg. 786 Example 1; and Lemma 3.3.1 in Weibel) that calculate $\operatorname{Ext}_{\mathbb Z}^i(A,B)$ for $i \geq 2$ using injective resolutions. I am wondering if there's a nice way to calculate $\operatorname{Ext}_{\mathbb Z}^i(A,B)$ using projective resolutions instead.

Answer 1

Because every subgroup of a free abelian group is free, every abelian group has a free (hence projective) resolution of the form $0 \to P_1 \to P_0$. So $\textrm{Ext}_\mathbb{Z}^i(A,B)=0$ when $i \geq 2$.