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In my current work I have to deal a lot with ext-groups (of modules). I feel kind of familar with the formalism, but I don't have a feeling about the meaning of ext.
Is there a informal/intuitive interpretation of ext-groups? I'm mostly interested in the case of $\mathcal{O}_X$-Modules for (toric) varieties or $\mathbb{C}[S]$-Algebras for a semi-groups $S$.

best regards, Johannes

edit:

ok, is see the connection bewteen group extensions of $N \to E \to M$ of $M,N$, but does this also hold for (f.g.) modules?

On the other hand, if I consider $Ext^i(M,N)$ by a free resolution $\cdots \to F_n \to \cdots \to F_1 \to F_0 \to M$ of $M$, does $Ext^i(M,N)$ tells me something about the morphisms in the i-th syzygy? e.g $Ext^1(M,N)$ 'are' the morphisms of the module generated by the relations of the generators of $M$ modulo the ones, which come from the trivial relations?

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    $\begingroup$ $n$-extensions. $\endgroup$
    – Fernando Muro
    Commented Dec 31, 2012 at 12:49
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    $\begingroup$ mathoverflow.net/questions/15016/about-higher-ext-in-r-mod $\endgroup$
    – M T
    Commented Dec 31, 2012 at 16:02
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    $\begingroup$ Not quite in line with your question. But if you were dealing with a reasonable topological space $X$, then $Ext$ groups of the constant sheaf with itself (in the category of constructible sheaves) are the cohomology groups of that space. More generally, extensions from the constant sheaf to any complex of sheaves is hypercohomology with coefficients in the complex. $\endgroup$
    – Reladenine Vakalwe
    Commented Dec 31, 2012 at 18:06
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    $\begingroup$ math.wayne.edu/~isaksen/Expository/carrying.pdf is a very soft introduction to ext, in terms of elementary school arithmetic and the "carrying" operation. Maybe you could think of it as being an unpacking of part of Fernando Muro's 1 letter and 1 word comment. $\endgroup$
    – Daniel Moskovich
    Commented Jan 1, 2013 at 4:16
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    $\begingroup$ After the edit, I think it's definitely clear that this question is not research level, is it? $\endgroup$
    – Fernando Muro
    Commented Jan 3, 2013 at 23:04

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I got an answer on stackexchange: https://math.stackexchange.com/questions/270228/geometric-intuitive-interpretation-of-ext.
thank you guys.

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