I would like to fill a detail in the answer given by Francesco Polizzi to the question "Is the dualizing sheaf on a Cohen-Macaulay scheme reflexive?".

Let $X$ be a normal, Cohen-Macaulay scheme of dimension $n$. Let $U$ be the smooth locus of $X$ and let $i : U \rightarrow X$ be the natural inclusion. Now, let $$ \omega_X = i_*{\Omega_U^n} $$ I would like to prove that $\omega_X$ coincides with the dualizing sheaf on $X$ in the sense of Serre duality. Basically, I would like to show for any locally free sheaf $F$ on $X$, we have $$ H^i(X, F) \cong H^{n-i}(X, F^\vee\otimes\omega_X)' $$

My immediate thought is to use projection formula for $F^\vee\otimes\omega_X$. Use projection formula, we have $$ H^{n-i}(X, F^\vee\otimes\omega_X) = H^{n-i}(X, i_*(F|_U\otimes\Omega_U^n)) $$ But I don't know how to get back to $H^i(X, F)$ from the right side of the above equation.