I am reading the paper *Frobenius splitting of Hilbert schemes
of points on surfaces* by Kumar and Thomsen. At the end of Lemma 11, they seem to imply that the dualizing sheaf on a Cohen-Macaulay scheme is reflexive. I am not familiar with reflexive sheaves, so I would like to ask whether this statement is true or not.



Is the dualizing sheaf on a Cohen-Macaulay scheme reflexive?
The answer is *yes*. Let me briefly explain why.


There are several ways to define a canonical sheaf on a projective variety $X$. One of them, that works for any *normal* $X$ is the following. Let $U$ be the smooth locus of $X$ and $i \colon U \hookrightarrow X$ be the natural inclusion. Then define $$\omega\_X:=i\_\*(\Omega^n\_U),$$
where $n:=\dim X$. As the sections of $\Omega^n\_U$ do not
depend on a subset of codimension $2$, neither do the sections of $\omega\_X$. In other words, $\omega\_X$ is a so-called *normal sheaf*. Moreover, by construction $\omega\_X$ is also torsion-free, so it is reflexive by the second characterization in [**1**, Proposition 1.6].


When $X$ is Cohen-Macaulay, this sheaf coincides with the dualizing sheaf in the sense of Serre duality. A different way to define a canonical sheaf for embedded varieties $X \subset \mathbb{P}^{N}$ is to consider $$\omega\_X' := h^{-n}(\omega\_X^{\bullet}),$$  

where $\omega\_X^{\bullet}$ is the so-called *dualizing complex*. Again, when $X$ is Cohen-Macaulay this construction provides a reflexive sheaf, that coincides with $\omega\_X$ outside a subset of codimension $\geq 2$. Then reflexivity implies $$\omega\_X'\simeq \omega\_X,$$
see [**2**, Section 5].


**References.**


**[1]$\,$** *Robin Hartshorne*, [**Stable reflexive sheaves**](http://dx.doi.org/10.1007/BF01467074), *Math. Ann.* **254** (1980), no. 2, 121--176. 


**[2] $\,$** *Sándor J. Kovács*, [**Singularities of stable varieties**](http://www.ams.org/mathscinet-getitem?mr=3184177), *Handbook of moduli. Vol. II* 159--203.


