I am trying to use Grothendieck duality (Duality) to prove that the dualising sheaf $\omega_X$ of a nodal curve $X$ can be described as the pushforward sheaf of the sheaf of differential forms on the normalization $\widetilde{X}$ with at most simple poles on the preimages of singularities such that the sum of their residues over the preimages of any singular point is zero(eg, here), but I met some difficulties as I now explain.
So assume I can embed $f:X\rightarrow P^2$, then by (4) of Duality we have $\omega_X^{\bullet}=f^{!}\omega_{P^2}^{\bullet}$. Assume further that $X$ is cut out by a single equation so that it is a complete intersection, by (8) of Duality we have $f^{!}\omega_{P^2}^{\bullet}=Lf^{*}(\omega_{P^2}^{\bullet})\otimes_{O_X}O_{P^2}(X)[-1]$, where $Lf^{*}$ is the derived pullback.
Now since $\omega_{P^2}^{\bullet}=\Omega^2_{P^2}[2]$ is an actual invertible sheaf sitting in degree 2, I assume I can regard it as K-flat, in a sense of (K-flat), and $Lf^{*}(\omega_{P^2}^{\bullet})=f^*(\omega_{P^2}^{\bullet})$.
Now my question is: how to describe the sheaf $f^*(\omega_{P^2}^{\bullet})$ when $X$ is not smooth(especially, how to describe it at the singular points), and how can I prove that $Lf^{*}(\omega_{P^2}^{\bullet})\otimes_{O_X}O_{P^2}(X)[-1]$ is actually the same sheaf with the sheaf I described in the first paragraph?
Any comments or helps are welcome!
