Let $k$ be an algebraically closed field and let $X$ be a Cohen-Macaulay variety over $k$, i.e. all local rings are Cohen-Macaulay (perhaps this can later be generalized). What is the dualizing sheaf on $X$, what is its central/characterizing property and can one write it down explicitly?

I'm asking this in such a naive way (knowing that there is a vast literature on this) because I have the following problem: I'm (obviously) not an expert in algebraic geometry and came across a definition stating that the dualizing sheaf of $X$ is $\omega_X := j_* \omega_{X_s}$, where $j:X_s \rightarrow X$ is the inclusion of the smooth locus $X_s$ of $X$ and $\omega_{X_s} := \mathrm{det}(\Omega_{X_s/k}^1)$. I can take this of course as a definition (it is pretty easy, and I also know something about dualizing sheaves on smooth varieties) but I don't know what the central properties of this sheaf in this setting are. I know that there is Grothendieck-Verdier-Neeman(-more names) duality and while browsing through Hartshorne's book Residues and Duality I tried to deduce this definition from the abstract "nonsense" but I failed. I know that there exists this very general duality and that there happens something in the Cohen-Macaulay case but this was over my head!

Amnon Neeman defines in his article Derived categories and Grothendieck duality a dualizing complex of a noetherian and separated scheme $X$ to be an object $\mathcal{J} \in \mathbf{D}^b( \mathrm{Coh}(X))$ such that $\mathbb{R}\mathcal{H}om(-,\mathcal{J}):\mathbf{D}^b(\mathrm{Coh}(X))^{op} \rightarrow \mathbf{D}^b(\mathrm{Coh}(X))$ is an (triangulated) equivalence. Is it correct that if $X$ is a (separated?) noetherian Cohen-Macaulay scheme the sheaf defined above is a dualizing complex in this sense and that this is the characterizing property I was asking for? This is the only idea I have so far... (Can I replace the separated by something that includes at least all affine varieties?)

Let $k$ be an algebraically closed field and let $X$ be a Cohen-Macaulay variety over $k$, i.e. all local rings are Cohen-Macaulay (perhaps this can later be generalized). What is the dualizing sheaf on $X$, what is its central/characterizing property and can one write it down explicitly?

I'm asking this in such a naive way (knowing that there is a vast literature on this) because I have the following problem: I'm (obviously) not an expert in algebraic geometry and came across a definition stating that the dualizing sheaf of $X$ is $\omega_X := j_* \omega_{X_s}$, where $j:X_s \rightarrow X$ is the inclusion of the smooth locus $X_s$ of $X$ and $\omega_{X_s} := \mathrm{det}(\Omega_{X_s/k}^1)$. I can take this of course as a definition (it is pretty easy, and I also know something about dualizing sheaves on smooth varieties) but I don't know what the central properties of this sheaf in this setting are. I know that there is Grothendieck-Verdier-Neeman(-more names) duality and while browsing through Hartshorne's book Residues and Duality I tried to deduce this definition from the abstract "nonsense" but I failed. I know that there exists this very general duality and that there happens something in the Cohen-Macaulay case but this was over my head!

Amnon Neeman defines in his article Derived categories and Grothendieck duality a dualizing complex of a noetherian and separated scheme $X$ to be an object $\mathcal{J} \in \mathbf{D}^b( \mathrm{Coh}(X))$ such that $\mathbb{R}\mathcal{H}om(-,\mathcal{J}):\mathbf{D}^b(\mathrm{Coh}(X))^{op} \rightarrow \mathbf{D}^b(\mathrm{Coh}(X))$ is an (triangulated) equivalence. Is it correct that if $X$ is a (separated?) noetherian Cohen-Macaulay scheme the sheaf defined above is a dualizing complex in this sense and that this is the characterizing property I was asking for? This is the only idea I have so far...