Comments on: M-geometry (1)

By: javier

javier — Fri, 21 Sep 2007 14:36:22 +0000

But, if the variety [tex]X[/tex] has a singularity at the point [tex]x[/tex], what is exactly the ‘tangent space’? I mean, the ‘tangent’ thing to a singularity is not a vector space at all, is it?

Btw, why the ‘M’ in ‘M-geometry’? Does it have any concrete meaning?

By: lieven

lieven — Mon, 17 Sep 2007 09:21:53 +0000

Javier, I think X being reduced is enough. In general if M is an n-dml A-representation and the variety rep(n) of all n-dml A-reps is reduced, then the Ext-space is the normal space in M to the GL(n)-orbit of M. Here, A is commutative, n=1 and so X=rep(1) and the GL(1)-action is trivial, so it is just the tangent space to X.

Edited a bit later : of course you do not need the general result in this case. If x is the point corresponding to the algebra map

[tex]\rho~:~\C[X] \rightarrow \C[/tex]

(evaluation in x) then the Ext-space is just the space of all [tex]\rho[/tex]-derivations and so coincides with the tangent space.

By: javier

javier — Mon, 17 Sep 2007 09:03:29 +0000

This may be well known, but I’ll ask anyway,
When you mention the equality
[tex]dim_{\C}~Ext^1_{\C[X]}(S_x,S_x) = dim_{\C}~T_x~X [/tex]
between the dimension of the [text]Ext[/tex] space and the tangent space in the classical setting. Is this equality always true, or do you require smoothnes of the variety [tex]X[/tex] at the point [tex]x[/tex].
In other words, can singularities be characterized by some property of the [tex]Ext[/tex] spaces?
Some other questions came while (re)reading this, but I’ll save them for later.

By: lieven

lieven — Sat, 15 Sep 2007 18:21:15 +0000

I guess the object you are interested in is the [tex]A_{\infty}[/tex] structure on the extension-algebra [tex]Ext^{\ast}_A(M,M)[/tex] for all semisimple A-modules M. A good paper on this is by Bernhard Keller “Introduction to Ainfinity algebras and modules” available via the arXiv (i think) or via his homepage.

Im mostly interested in algebras where the first 2 terms of the approximation scheme are enough, so called formally smooth algebras or quasi-free algebras, so in my cases the quiver is all I need but I know it can be extended…

A final comment about TeX. You can use TeX in comments but instead of $latex as opening tag use [ tex ] (without the spaces) and the closing $ should be replaced by [ /tex ] (sorry about that). (Ive edited your comment to get the TeX properly displayed, in fact better than mine because of the gray background…)

By: Greg

Greg — Sat, 15 Sep 2007 18:01:45 +0000

It seems like this is the ‘first-order’ approximation to the representation theory of [tex]A[/tex], in the way that the K-theory is the ‘zeroth-order’ approximation.

What I mean is, [tex]K(A)[/tex] throws aways all the representation theory info in the [tex]Ext[/tex]s, whereas the tangent quiver seems to throw away all the representation theory info in [tex]Ext^n[/tex], [tex]n > 1[/tex]. Are there similar constructions that yield algebras that measure [tex]Ext^n[/tex], for [tex]n[/tex] greater than some integer? And, if so, can they form a directed system with a defined limit, some sort of ‘pro-finite’ approximation of the original algebra?