Why is this the dualising sheaf of a singular curve?

Question

If $X$ is a curve with a nodal singularity at $x$, it's referred to here and here that its dualising sheaf is $$\omega_X \ = \ \pi_*(\Omega_{X}(p_1+\cdots+p_n)').$$ Here, $\pi:X\to X'$ is the normalisation and $\Omega_{X}(p_1+\cdots+p_n)'$ are the forms $\theta$ which have at worst simple poles at $\pi^{-1}(x)=\{p_1,...,p_n\}$, satisfying $$\sum_i \text{Res}_{p_i}\theta\ = \ 0 .$$

1. What is the proof that this is the dualising sheaf? I can't find a reference, even in the complex-analytic case.
2. What happens when $X$ has other types of singularities (and what is the proof)? For instance, working complex-analytically, what if $X$ is locally $$(z-\alpha_1w)\cdots(z-\alpha_nw)\ = 0 \ ?$$

Answer 1

There are two ways to compute the dualising sheaf of a curve:

1. Mimick Serre's proof of Serre duality (p.8-13 of this link) for nonsingular curves, which directly gives the result in the question and explains where the residue map comes from (it comes from the residue pairing which you need to prove Serre duality). Note that it's quite easy to compute.
2. To compute $\omega_X$ even faster, use that if $X\subseteq \mathbb{P}^n_k$ is a codimension $r$ closed subscheme, then $\omega_X=\mathcal{E}\text{xt}^r_{\mathbb{P}^n}(\mathcal{O}_X,\Omega^n_{\mathbb{P}^n})$ is its dualising sheaf (Hartshorne III,7). If $X$ is the vanishing set of a degree $d$ polynomial $f$ for instance, take a resolution of $\mathcal{O}_X$ by projective $\mathcal{O}_{\mathbb{P}^n}$-modules: $$0 \ \longrightarrow \ \mathcal{O}_{\mathbb{P}^n}(d) \ \stackrel{\cdot f}{\longrightarrow} \ \mathcal{O}_{\mathbb{P}^n} \ \longrightarrow \ \mathcal{O}_X$$ apply $\mathcal{H}\text{om}_{\mathbb{P}^n}(-,\Omega^n_{\mathbb{P}^n})$ and take the first cohomology.