Algebraic Geometry – Is the Dualizing Sheaf on a Cohen-Macaulay Scheme Reflexive?

Question

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.

Answer 1

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, Math. Ann. 254 (1980), no. 2, 121--176.

[2] $\,$ Sándor J. Kovács, Singularities of stable varieties, Handbook of moduli. Vol. II 159--203.