Hi!
Given $Z$ a subscheme of a smooth variety $X$ over an algebraically closed field. Is there a way to determine if it is or not reduced by examining the ext sheaves $\mathrm{Ext}^i(O\_Z,\omega\_X)$, where $\omega\_X$ is the dualizing sheaf on $X$? I will be more precise: suppose that I know that there exists $d$ such that $\mathrm{Ext}^i(O\_Z,\omega\_X)=0$ if $i>d$ and $\mathrm{Ext}^d(O\_Z,\omega\_X)$ is a skyscraper sheaf over some point $p\in X$ (of the other ext-sheaves I know nothing). Can I deduce something about the reduceness of (some component of) $Z$? I feel some what that if I have some component that passes from the point $p$ than it cannot be "double", but I really do not know how to check it...
thank you very much



Non reduced subschemes and ext sheaves
I just want to clarify, the $Ext$ you are considering are Sheaf-Ext, not global section Ext, right?


Regardless, I think the answer is probably you can't determine that just from vanishing / support. Consider for example $X = \text{Spec } k[x, y]$ and $Z = V(x)$ and $Z'= V(x^2)$. They have the same vanishing behavior for Ext, but one is reduced and the other is not. There are some ways in which a scheme fails to be reduced which can be detected by $Ext$ though, for example $k[x,y]/(x^2, xy)$ has a more complicated series of $Ext$s. 


What you can detect is the (non-)Cohen-Macaulay'ness (or more generally, facts about the depth) of $Z$. See for example, Corollary 3.5.11 in Bruns-Herzog *Cohen-Macaulay rings*. Basically, if all the exts that can vanish do vanish, then you are Cohen-Macaulay. 


There are different ways a variety can be non-reduced. It can have associated points that are not generic points, ie the ring can have associated primes which are not minimal. 


Non-reducedness in the form of having these non-generic associated points is an obstruction to being Cohen-Macaulay. In particular, Cohen-Macaulay rings are unmixed. Therefore, those obstructions can be found in Ext-groups, but there can be other reasons that a ring is not Cohen-Macaulay. Therefore, other things can be seen in the Ext-groups as well.


Of course, at some level all the Ext groups put in a complex, $R Hom\_X(O\_Z, \omega\_X)$ isn't really any different data than $O\_Z$ itself (apply $R Hom\_X( - , \omega\_X)$ again and you get $O\_Z$ back). Thus reducedness can be seen from that complex. But I don't think you can see it just from vanishing / support.


