Non reduced subschemes and ext sheaves

Question

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

Answer 1

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.

Answer 2

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 non-minimal associated primes", so you can see it in the Ext-groups as Karl explains. As far as I know, $R_0$ can't be thought of as a condition about Ext-groups, but it is often not hard to check, since you only need to check it at the generic points of $X$.

Answer 3

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 $j>d-t$ and $\mathscr Ext^{d-t}_p(\mathscr O_X,\omega_X^\bullet[-d])\neq 0$ 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 Cohen-Macaluay (CM) you might as well look this up. Being CM is equivalent to this complex being quasi-isomorphic 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 non-reduced schemes (e.g., any non-reduced 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.