Dualizing sheaf of nodal genus two curve I'm interested in extending Riemann-Roch and Serre duality to bundles or sheaves defined on nodal curves. The extension of Riemann-Roch I think is straightforward: simply replace the geometric genus by the arithmetic genus. Unless I'm mistaken, Serre duality can be extended to nodal curves by replacing the canonical bundle with the _dualizing sheaf_. I've never worked with a dualizing sheaf concretely, so I was hoping someone could help me in one very explicit example. Let $C$ be the nodal curve given as the glueing of two elliptic curves at their origins. One can think of this as a particular kind of genus two curve. Obviously, the normalization is just the two disjoint elliptic curves. But what exactly is the dualizing sheaf $\omega_{C}$ and how does one carry out the computation?

First, simple nodal curves are locally planar and so the dualizing sheaf is a line bundle. So, if the elliptic curves are $E_1,E_2$, one has an exact sequence $0\to \mathcal{O}_C\to \mathcal{O}_{E_1}\oplus\mathcal{O}_{E_2}\to k(p)\to 0$, where $p$ is the origin. Dulaizing, you get, $0\to\mathcal{O}_{E_1}\oplus\mathcal{O}_{E_2}\to \omega_C\to k(p)\to 0$. Most calculations you need can be made using this sequence.