Let $X$ be a smooth projective variety of dimension $d$ over $\mathbb C$ and let $E$ be a zero-dimensional coherent sheaf on $X$. The dual sheaf $E^D=\mathscr Ext\_X^d(E,\omega\_X)$ is again zero-dimensional.



> 
> *Question*. Do $E$ and $E^D$ have support of the same length?
> 
> 
> 


In other words, if $Z,Z^D\subset X$ are the supports of $E,E^D$ respectively, is it true that $h^0(X,\mathscr O\_Z)=h^0(X,\mathscr O\_{Z^D})$? A more ambitious question would be to ask if the supporting points are the same. Maybe these questions can be reduced to a commutative algebra problem involving canonical modules, but I do not see how to do that. Any reference (I feel this is well-known) is very much appreciated. Thanks!



Support of 0-dimensional sheaf and its dual
I misread your question, so the comments above are answering a different question than you asked. I thought you were asking about the lengths of the sheaf and its dual sheaf, not the lengths of the supports. At any rate, this does follow from duality, since the dual of the dual equals the original module (Proposition 5.1, p. 275 of "Residues and Duality"). 


Let us work locally: so $R$ is a local, Noetherian, Gorenstein $k$-algebra of dimension $d$ with residue field $k$ (so $k$ is a coefficient field). Denote a dualizing module by $\omega\_R \cong R$. Let $E$ be a finite length $R$-module. Let $f\in R$ be any element in the annihilator of $M$. Then since the multiplication by $f$ homomorphism, $$L\_{E,f}:E \to E, \ m \mapsto fm,$$ is the zero homomorphism, the same is true on $E^D = \text{Ext}^d\_R(E,\omega\_R)$, since Ext is an additive functor. Thus, the annihilator ideal of $E$ is contained in the annihilator ideal of $E^D$. On the other hand, $(E^D)^D$ is ~~congruent~~ isomorphic to $E$ as a finite length $R$-module. Thus, also the annihilator ideal of $E^D$ is contained in the annihilator ideal of $E$. So the two annihilator ideals are equal, i.e., the scheme-theoretic support of $E$ equals the scheme-theoretic support of $E^D$ (although $E$ and $E^D$ need not be isomorphic as $R$-modules).


