(geometric/intuitive) interpretation of ext - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T03:36:10Z http://mathoverflow.net/feeds/question/117723 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/117723/geometric-intuitive-interpretation-of-ext (geometric/intuitive) interpretation of ext Johannes 2012-12-31T12:42:44Z 2013-01-04T18:03:03Z <p>Hi folkz,<br> 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.<br> 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 $S$.</p> <p>best regards, Johannes</p> <p>edit:</p> <p>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? </p> <p>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?</p> http://mathoverflow.net/questions/117723/geometric-intuitive-interpretation-of-ext/118073#118073 Answer by Johannes for (geometric/intuitive) interpretation of ext Johannes 2013-01-04T18:03:03Z 2013-01-04T18:03:03Z <p>I got an answer on stackexchange.<br> thank you guys.</p>