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I am working on extensions in general but for sake of simplicity we can assume it's a module here. I am interested in an extension of the form $$0\to B \to E_2\to E_1 \to A \to 0$$ which is an extension of $A$ by $B$. I want to find an example of this that is relatively "simple", I tried some modulos like $\mathbb{Z}_3$ by $\mathbb{Z}_3$ but that didn't work out as I realized it doesn't have it, if I did it correctly.

Are there any good simple module examples like that to illustrate such an extension?

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  • $\begingroup$ If I understand correctly, you want this sequence to represent a non-trivial element of $Ext_R^2(A,B)$? $\endgroup$ – Ben Mar 4 '16 at 18:37
  • $\begingroup$ Trivial and non-trivial as I cannot think of either. $\endgroup$ – Zelos Malum Mar 4 '16 at 18:38
  • $\begingroup$ The sequence $B \overset{=}\to B \overset{0}\to A \overset{=}\to A$ is always trivial. I can explain how to get non-trivial examples in an answer. $\endgroup$ – Ben Mar 4 '16 at 18:41
  • $\begingroup$ I would like one that is NOT equivalent in some manner to just a short exact sequence, please. $\endgroup$ – Zelos Malum Mar 4 '16 at 18:42
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Here is one example. Let $R = k[x]/x^2$ and $A = B = k$ with $xA = xB = 0$. Then we have a free resolution $\cdots \to R \overset{x} \to R \overset{x}\to R \to A$, hence $Ext_R^2(A,B) = k$ generated by the quotient map $R \to k$. The corresponding extension is: $$k \overset{x}\to R \overset{x}\to R \to k.$$ Similar extensions generate the higher Exts as well.

Note: This example is nice because you can quickly calculate the algebra structure $Ext_R(k,k) \cong k[t]$.

More generally, if you take a projective resolution $$\cdots \to P_3 \overset{f}\to P_2 \to P_1 \to P_0 \to A\to 0$$ and truncate it to $$0\to B \to P_1 \to P_0 \to A\to 0$$ where $B = \text{coker}(f)$, this will give an element of $Ext^2(A,B)$ which is trivial if and only if $B \to P_1$ is a split injection. Being a split injection means $P_1/B$ is projective, so that $A$ has a projective resolution of length 1 given by $P_1/B \to P_0 \to A$, in particular $A$ has projective dimension at most 1.

If $R$ is a regular local ring, then the projective dimension of the residue field $R/\mathfrak m$ equals the Krull dimension of $R$. Thus taking a regular local ring of dimension at least 2 and using the natural Koszul resolution will give you many more examples.

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