## EXT for abelian group [closed]

Let $A$ and $B$ are abelian groups. Find $Ext_{\mathbb{Z}}^{n}(A,B)$. (For all $n$)

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Where's the question? I just see a command... – Steve D Dec 13 2010 at 5:36
mathoverflow.net/faq#whatnot Where does this question originate? Which examples have you already tried? – Yemon Choi Dec 13 2010 at 5:39
@Jason How do you know that the downvoter hasn't already provided an explanation? E.g. upvoting a comment that gives an explanation is one of the options of doing so (no point in repeating something twice in comments). – Alex Bartel Dec 13 2010 at 6:04
One remark: if $n > 1$ then the Ext group vanishes (no matter what $A$ and $B$ are). – Emerton Dec 13 2010 at 6:19
As Emerton pointed out, it is only a question of calculating $Ext^1$. This is a must-do (easy) exercise for cyclic $A$ and $B$. In Cartan-Eilenberg you will find theorems about the behavior of Ext under injective and projective limits, which will let you dispose of lots of cases. For example, $Ext^1(A,B)$ is iso to $Hom(A,B)$ (but not naturally isomorphic) when $A$ and $B$ are finite. The realm of all abelian groups is not to be taken lightly; see the two-volume book on the subject by Fuchs, and this can get interesting. – Amritanshu Prasad Dec 14 2010 at 5:16