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I have been studying modules and homological algebra as of late but somehow I have missed how to calculate Hom(A,B) for abelian groups, modules and Hom(A,_)/Hom(_,B) for exact sequences. I have no problem with the abstract nonsense in the subject. But explicit examples are useful and this construction is seldom mentioned explicitly in texts so I suspect it's pretty trivial but I'd still like someone to write this out for me.

Is it as as simple as finding generators in A and picking elements in B to send them to? (with careful adjectives of course, but I am looking for the general idea.)

Also some examples would be really helpful! (If they show a general idea, not just some clever construction.)

Thanks!

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    Yes, and then making sure all the relations are satisfied. Any specific examples you're having trouble with?2011-08-02
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    Say for example all combinations of $\mathbb{Z}$, $\mathbb{Z}_p$ and $\mathbb{Q}$ as abelian groups. Not really difficult stuff, but I had an exam where calculating these took me very long and was error-prone, thus the question.2011-08-02
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    by $\mathbb{Z}_p$ do you mean the additive group of the $p$-adic integers or $\mathbb{Z}/p\mathbb{Z}$?2011-08-02
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    Oh, just $\mathbb{Z}/p\mathbb{Z}$2011-08-02
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    How you compute depends on how the groups A, B, &c. are given. I think it would be best if you proposed a concrete example, and then hopefully we can show you how to handle it. The general question «how does one compute homs between abelian groups?» has no useful answer.2011-08-02
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    see the above comment, though I suspect Qiaochu will give a good answer (as always!). That the general question has no useful answer is a shame though, maybe that's why I couldn't find it!2011-08-02
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    For finitely generated modules over PIDs (including the case of finitely generated abelian groups) the universal properties of the direct sum/product yield that the problem of computing homs can be reduced to the problem of computing $\mathrm{Hom}(C,D)$, where $C$ and $D$ are cyclic; for f.g. abelian groups, this essentially solves the problem for this subclass.2011-08-02

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