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 extsheaves 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
I just want to clarify, the $Ext$ you are considering are SheafExt, 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)CohenMacaulay'ness (or more generally, facts about the depth) of $Z$. See for example, Corollary 3.5.11 in BrunsHerzog CohenMacaulay rings. Basically, if all the exts that can vanish do vanish, then you are CohenMacaulay.
There are different ways a variety can be nonreduced. It can have associated points that are not generic points, ie the ring can have associated primes which are not minimal.
Nonreducedness in the form of having these nongeneric associated points is an obstruction to being CohenMacaulay. In particular, CohenMacaulay rings are unmixed. Therefore, those obstructions can be found in Extgroups, but there can be other reasons that a ring is not CohenMacaulay. Therefore, other things can be seen in the Extgroups 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.

$\begingroup$ Thank you very much for your answer! I will check the reference about cohen maculay rings that you suggested! The fact is that the subscheme $Z$ I am working with is very simple: set theoretically is a union of very nice smooth varieties (i.e complex tori). The intersection points between different components are not a big deal, what I would like to check is that the settheoretic description holds also schemetheoretically (or that at least one of the complex tori is not "doubled"). thak you again for tuor time! $\endgroup$ – Rurik Jun 9 '11 at 11:49
To complement Karl's answer, one should mention the following nice result.
A Noetherian commutative ring is reduced if and only if it is $R_0$ and $S_1$.
See, for example, Theorem 4.5.2 in Huneke and Swanson.
The condition $S_1$ is equivalent to "has no nonminimal associated primes", so you can see it in the Extgroups as Karl explains. As far as I know, $R_0$ can't be thought of as a condition about Extgroups, but it is often not hard to check, since you only need to check it at the generic points of $X$.
It is probably worth looking at this from another point of view and ask what your condition means in general:
Theorem (a variant of Grothendieck's vanishing for local cohomology) Let $p\in X$ be a closed point on a noetherian scheme. Let $d=\dim X$ and $t=\mathrm{depth}_xX$. Then $\mathscr Ext^j_p(\mathscr O_X,\omega_X^\bullet[d])=0$ for $j>dt$ and $\mathscr Ext^{dt}_p(\mathscr O_X,\omega_X^\bullet[d])\neq 0$ where $\omega_X[d]$ is the dualizing complex shifted by $d$.
From your answer I assume that the words "dualizing complex shifted by $d$" might not mean much, but if you are looking up CohenMacaluay (CM) you might as well look this up. Being CM is equivalent to this complex being quasiisomorphic to the sheaf $\omega_X$.
This implies that if $X$ is CM, then $\mathscr Ext^j_p(\mathscr O_X,\omega_X)=0$ for all $j>0$, and hence knowing this alone will not give you what you want, as there are CM but nonreduced schemes (e.g., any nonreduced hypersurface). Then again, as David mentioned, if you know that your scheme is generically reduced, then being reduced everywhere is indeed a depth condition.
If $X$ is not CM, then your condition is not equivalent to the condition in this theorem, but they are related. In particular, there is a spectral sequence computing $\mathscr Ext^j_p(\mathscr O_X,\omega_X^\bullet[d])$ where $\mathscr Ext^j_p(\mathscr O_X,\omega_X)$ appears on one of the border lines. I'm afraid it gets a little technical soon, but having looked at $\mathscr Ext$'s already suggests that you're not afraid of that.