Hi folkz,
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?

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?

Hi folkz,
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

Hi folkz,
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?