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Can you please give some examples of computation of the derived functors $\operatorname{Tor}_1$ and $\operatorname{Ext}^1$ for some simple cases, say $R=\mathbb{Z}$ or $R=\mathbb{Z}[G]$ for some finite group $G$?

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    http://www.math.wichita.edu/~pparker/classes/handout/torext.pdf lists basic properties and some examples.2012-02-12
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    All textbooks about homological algebra that I know of include examples. Which have you looked?2012-02-12

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